Set-Valued Graphs - ispacs

Available online at www.ispacs.com/jfsva Volume 2012, Year 2012 Article ID jfsva-00127, 17 pages doi:10.5899/2012/jfsva-00127 Research Article

Set-Valued Graphs Kumar Abhishek 1∗, Germina K. Agustine

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(1) Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641 105, India. (2) Mary Matha Arts & Science College, Mananthavady - 670 645, India.

c Copyright 2012 ⃝Kumar Abhishek and Germina K. A. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract A set-indexer [3] of a given graph G = (V, E) is an assignment f of distinct nonempty subsets of a finite nonempty ‘ground set’ X of cardinality n to the vertices of G so that the values f ⊕ (e), e = uv ∈ E, obtained as the symmetric differences f (u)⊕f (v) of the subsets f (u) and f (v) of X, are all distinct. A set-indexer f of a graph G, is called a segregation of X on G if the sets f (V (G)) = {f (u) : u ∈ V (G)} and f ⊕ (E(G)) = {f ⊕ (e) : e ∈ E(G)} are disjoint, and if, in addition, their union is the set Y (X) = P(X) − {∅} of all the nonempty subsets of X where P(X) denotes the power set of X, then f is called a setsequential labeling of G. A graph is hence called set-sequential if it admits a set-sequential labeling with respect to some ‘ground set’ X. A set-indexer f of a (p, q)-graph G = (V, E) is called a set-graceful labeling [3] of G if there exists nonempty ground set X such that f ⊕ (E) = P(X) − {∅} and G is set-graceful if it admits a set-graceful labeling. In this report we provide a complete characterization of set-sequential caterpillar of diameter four. We also a provide a new necessary condition for a graph to be set-sequential. Keywords : Graphs, Set-indexer, Set-sequential labeling, Set-graceful labeling.

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Introduction

For all terminology and notation in graph theory, not defined specifically in this paper, we refer the reader to Harary [9]. Unless mentioned otherwise, all the graphs considered in this note are simple, loop-free and finite. Acharya introduced to the notion of set-valuations of graphs [3], as follows: A setindexer of a given graph G = (V, E) is an assignment f of distinct nonempty subsets of a ∗

Email address: [email protected]

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finite nonempty ‘ground set’ X = {x1 , x2 , . . . , xn } of cardinality n to the vertices of G so that the values f ⊕ (e), e = uv ∈ E, obtained as the symmetric differences f (u) ⊕ f (v) of the subsets f (u) and f (v) of X, are all distinct. In [3], it is proved that every graph admits a set-indexer. In the recent literature the notion of set-valuation has been surveyed by Hegde [15] using the term set-coloring. However we would follow the terminology used by Acharya in [3]. Acharya and Hegde [6] viewed the notion of set-sequential graphs as a set analogue of the well known sequential graphs, which was independently introduced and studied by Acharya [4] and Slater [13], as follows: A set-indexer f of a graph G, is called a segregation of X on G if the sets f (V (G)) = {f (u) : u ∈ V (G)} and f ⊕ (E(G)) = {f ⊕ (e) : e ∈ E(G)} are disjoint, and if, in addition, their union is the set Y (X) = P(X) − {∅} of all the nonempty subsets of X then f is called a set-sequential labeling [3] of G. A graph is hence called set-sequential if it admits a set-sequential labeling with respect to some ‘ground set’ X. Acharya [3] viewed the notion of set-graceful graphs as a set analogue of the well known graceful graphs, which was introduced by Rosa [2], as follows: A setindexer(set coloring [15]) f of a (p, q)-graph G = (V, E) is called a set-graceful labeling [3] of G if there exists nonempty ground set X such that f ⊕ (E) = P(X) − {∅} and G is set-graceful if it admits a set-graceful labeling. In [6], it has been shown that for any (p, q)-graph G, the condition that p + q = 2n − 1 for some positive integer n is necessary for G to admit a set-sequential labeling and the condition that q = 2n − 1 for some positive integer n is necessary for G to admit a setgraceful labeling; we will call these condition the ‘Acharya-Hegde necessary conditions’. It has also been observed that, in general, this necessary condition for a (p, q)-graph to admit a set-sequential labeling is not sufficient as, for example, P4 , the path of order 4, satisfies this necessary condition but the graph is not set-sequential. Further, the following two conjectures were respectively made in [6] and [15]. Conjecture 1.1. [6]

No path Pn , n > 2 is set-sequential.

Conjecture 1.2. [15] The complete bipartite graph Ka,b is set-graceful if and only if it is a star with a = 1 and b = 2n−1 . The authors have learnt that Conjecture 1.1 has been disproved [1] and Conjecture 1.2 has been recently settled by Balister, et.al, [16] . The following is a useful link between the notion of set-sequential and set-graceful graphs. Theorem 1.1. [3] A graph G is set-sequential if and only if G + K1 with V (K1 ) = {v} has a set-graceful labeling f such that f (v) = ∅. Following Theorems were respectively proved in [7] and [15]. Theorem 1.2.

Almost all graphs are not set-graceful.

Theorem 1.3.

If G(p > 2) has:

1. exactly one or two vertices of even degree or 2. exactly three vertices of even degree at least two of which are adjacent or 3. exactly four vertices of even degree, say v1 , v2 , v3 , v4 such that v1 v2 and v3 v4 are edges in G, then G is not set-sequential. 2

Owing to the Acharya-Hegde necessary conditions, not every graph admits a setsequential (set-graceful) labeling and in view of Theorem 1.1 and Theorem 1.2 its is quite apparent that “Almost all graphs are not set-sequential ”. So far there is no characterization of set-sequential ( set-graceful) graphs. Hence, while obtaining in general a ‘good’ characterization of a set-sequential (set-graceful) graphs appears a formidable open problem [15] it becomes imperative to recognize graphs which are set-sequential (set-graceful). In particular, the problem of characterizing set-sequential trees was raised in [5] and [8] and was studied in [16] for the case of binary trees. The first result in the next section gives a new necessary condition for a graph to be set-sequential followed by a complete characterization of set-sequential caterpillar of diameter four. We also identify and provide a complete characterization of a class of set-sequential tree of diameter four.

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New results

Theorem 2.1. Let G be a set-sequential graph with respect to a set X. Then, for every edge ab ∈ E(G) with d(a) = d(b) one has d(a) + d(b) < 2n−1 − 1 unless G ∼ = K5 . Proof. Suppose the theorem is false. Then, there exists a (p, q)-graph G having a setsequential labeling f with respect to a set X having an edge ab with d(a) = d(b), yet d(a) + d(b) ≥ 2n−1 − 1. Firstly, it is not difficult to verify that no graph with at most four vertices satisfies all the given properties, viz., a set-sequential graph H having an edge ab with d(a) = d(b) and d(a) + d(b) ≥ 2n−1 − 1, where n is the cardinality of the ground set X with respect to which there is a set-sequential labeling of H. Therefore, G has at least five vertices whence, by invoking the Acharya-Hegde necessary condition for a graph to be set-sequential, we see that n ≥ 3. If n = 3, it may be verified that q ≤ 2. Further, no graph on at least five vertices and at most two edges is set-sequential. Therefore, n ≥ 4. If n > 4 then it is not difficult to verify that p would never be an integer solution to the quadratic inequality p2 + p + 2 − 2n+1 ≥ 0 . Therefore, n = 4 whence which is set up by the facts that p + q = 2n − 1 and q ≤ p(p−1) 2 n we get p = 5. Since G is set-sequential, we have p + q = 2 − 1 which, when p = 5 and n = 4, yields that q = 10 and hence G ∼ = K5 and we know that it is set-sequential [6]. A caterpillar is a tree with the property that the removal of its endpoints leaves a path, we characterize set-sequential caterpillars of diameter four. Set-sequential caterpillars of diameter five and six has been characterized in [10]. But before we proceed any further we make the following conventions: by T [a, b, c] we would mean a caterpillar whose underlying path is P = [u, v, w] and u, v, w have degrees a, b, c in T [a, b, c]. For example T [2, 2, 2] is a path of length four or T [3, 3, 3] is as shown in the Fig. 1. Further we denote the pendant vertices adjacent to u, v, and w respectively as ui where 1 ≤ i ≤ a − 1, vj where 1 ≤ j ≤ b − 2 and wk where 1 ≤ k ≤ c − 1.

Note that there is no set sequential tree of diameter ≥ 4 with respect to a set X of cardinality three, as for if T is a tree of order ≥ 4 then |V (T )| + |E(T )| ≥ 9 where as 23 − 1 = 7. 3

u

v

Figure 1: Proposition 2.1. Proof.

w

T [3, 3, 3].

If T [a, b, c] is set-sequential then a, b, c are all odd > 1.

The proof follows from Theorem 1.3.

As the Proposition 2.1 suggests all the internal vertices of set-sequential caterpillar of diameter four are odd, we would require the following terminology and notions which would smoothen our presentation. Let So (n, 3) denote the set of all the 3-decompositions of an integer n into odd parts > 1. Where decomposition [11] is an order dependent partition of an integer n. For example Example 2.1. So (17, 3) = {(11, 3, 3), (3, 11, 3), (3, 3, 11), (9, 5, 3), (9, 3, 5), (5, 9, 3), (5, 3, 9), (3, 9, 5), (3, 5, 9), (7, 7, 3), (7, 3, 7), (3, 7, 7), (7, 5, 5), (5, 7, 5), (5, 5, 7)}. We say that So (n, 3) generates So (m, 3), m > n if each (x1 , x2 , x3 ) ∈ So (m, 3) can be expressed as (x1 , x2 , x3 ) = (y1 , y2 , y3 ) + (2r , 0, 0), or = (y1 , y2 , y3 ) + (0, 2r , 0), or = (y1 , y2 , y3 ) + (0, 0, 2r ), or = (y1 , y2 , y3 ) + (2r−2 , 2r−2 , 0), or = (y1 , y2 , y3 ) + (2r−2 , 0, 2r−2 ), or = (y1 , y2 , y3 ) + (0, 2r−2 , 2r−2 ), or = (y1 , y2 , y3 ) + (2r−3 , 2r−3 , 2r−2 ), or = (y1 , y2 , y3 ) + (2r−3 , 2r−2 , 2r−3 ), or = (y1 , y2 , y3 ) + (2r−2 , 2r−3 , 2r−3 ) for some r > 0 and some (y1 , y2 , y3 ) ∈ So (n, 3). Proposition 2.2.

So (2n+3 + 1, 3) is generated by So (2n+2 + 1, 3) for r = n + 2, n ≥ 1.

Proof. Let n ≥ 1 and let (x1 , x2 , x3 ) ∈ So (2n+3 + 1, 3) be a partition of 2n+3 + 1 into three odd parts, each part being at least 3. Without loss of generality assume that x1 ≥ x2 ≥ x3 . Note that the difference between 2n+3 + 1 and 2n+2 + 1 is 2n+2 . Our objective is to show that there exists some (y1 , y2 , y3 ) ∈ So (2n+2 + 1, 3) such that the difference (x1 , x2 , x3 ) − (y1 , y2 , y3 ) is a 3-tuple whose components add up to 2n+2 and whose components are zero or a power of 2 as in one of the nine types given above. We consider two cases, depending on the value of the largest odd part x1 :

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Case 1: Suppose x1 ≥ 2n+2 + 3: Then we set y1 = x1 − 2n+2 , y2 = x2 , y3 = x3 . The condition on x1 implies that y1 ≥ 3, and so we have the 3-tuple (y1 , y2 , y3 ) ∈ So (2n+2 +1, 3). Case 2: Suppose x1 ≤ 2n+2 + 1: Consider two subcases, depending on the value of x2 : Subcase a: Suppose x2 ≥ 2n+1 + 3. Then, x1 ≥ x2 ≥ 2n+1 + 3. So, we can set (y1 , y2 , y3 ) to be (x1 − 2n+1 , x2 − 2n+1 , x3 ), which is a partition of 2n+2 + 1 into three odd parts, each being of size at least 3. Subcase b: Suppose x2 ≤ 2n+1 + 1. Then, since x1 ≤ 2n+2 + 1 and x2 ≤ 2n+1 + 1, we have that x3 ≥ 2n+3 + 1 − (2n+2 + 1 + 2n+1 + 1) = 2n+3 − 2n+2 − 2n+1 − 1 ≥ 2n + 3. Hence, x2 ≥ 2n + 3. Also, x2 , x3 ≤ 2n+1 + 1 imply that x1 ≥ 2n+3 + 1 − (2n+1 + 1 + 2n+1 + 1) ≥ 2n+2 − 1 ≥ 2n+1 + 3. Hence, we can subtract from (x1 , x2 , x3 ) the 3-tuple (2n+1 , 2n , 2n ) to get the desired (y1 , y2 , y3 ) ∈ So (2n+2 + 1, 3).

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Theorem 2.2. some n ≥ 1.

T [a, b, c] is set-sequential if and only if (a, b, c) ∈ So (2n+2 + 1, 3) for

Proof. Let T [a, b, c] be set-sequential, then in view of Acharya-Hegde necessary conditions, the fact that there is no set-sequential tree of diameter four with respect to a set of cardinality three and Proposition 2.1, it is seen that (a, b, c) ∈ So (2n+2 + 1, 3) for some n ≥ 1. Conversely, let T [a, b, c] be such that (a, b, c) ∈ So (2n+2 + 1, 3) for some n ≥ 1. We shall show that T [a, b, c] is set-sequential. Let u, v, w be the internal vertices of T [a, b, c] and the pendant vertices adjacent to u, v, and w respectively be denoted as ui where 1 ≤ i ≤ a − 1, vj where 1 ≤ j ≤ b − 2 and wk where 1 ≤ k ≤ c − 1. Since, d(u) + d(v) + d(w) = a + b + c = 2n+2 + 1 it is easily seen that T [a, b, c] is of the order 2n+2 , and the number of pendant vertices are 2n+2 − 3. For n = 1, So (9, 3) = {(3, 3, 3)}. Thus T0 [3, 3, 3] is unique caterpillar of diameter four with 21+2 − 3 = 5 pendant vertices and is seen to be set-sequential with respect to with respect to X1 = {x1 , x2 , x3 } ∪ {x4 } of cardinality 3 + 1 = 4, as shown in Table 1. f1 V (T0 [3, 3, 3]) u, v, w ui , 1 ≤ i ≤ 2 vj , j = 1 wk , 1 ≤ k ≤ 2

: V (T0 [3, 3, 3]) → P(X1 ) − {∅} f1 (V (T0 [3, 3, 3]) f1 (u) = {x1 }, f1 (v) = {x2 }, f1 (w) = {x3 } f1 (u1 ) = {x2 , x3 , x4 }, f1 (u2 ) = {x1 , x2 , x4 } f1 (v1 ) = {x1 , x2 , x3 } f1 (w1 ) = {x4 } , f1 (w2 ) = {x1 , x3 , x4 }

Table 1:

Set-sequential labeling of T0 [3, 3, 3].

For n = 2, So (17, 3) is shown in example 2.1 and for (a, b, c) ∈ So (17, 3), the non isomorphic caterpillars of diameter four shown in the Table 2 and each of them contains 22+2 − 3 = 13 pendant vertices. (a, b, c) ∈ So (17, 3) (3, 3, 11) or (11, 3, 3) (3, 11, 3) (9, 5, 3) or (3, 5, 9) (5, 9, 3) or (3, 9, 5) (9, 3, 5) or (5, 3, 9) (3, 7, 7) or (7, 7, 3) (7, 3, 7) (5, 5, 7) or (7, 5, 5) (5, 7, 5)

Table 2:

All non isomorphic caterpillars T [a, b, c] such that (a, b, c) ∈ So (17, 3). T1 [3, 3, 11], T2 [3, 11, 3], T3 [3, 5, 9], T4 [5, 9, 3], T5 [9, 3, 5], T6 [3, 7, 7], T7 [7, 3, 7], T8 [5, 5, 7], T9 [5, 7, 5].

Non isomorphic caterpillars T [a, b, c] associated with (a, b, c) ∈ So (17, 3).

And, each of Ti [a, b, c], 1 ≤ i ≤ 9 such that (a, b, c) ∈ So (17, 3) seen to be set-sequential with respect to X2 = X1 ∪ {x5 } = {x1 , x2 , x3 } ∪ {x4 } ∪ {x5 } of cardinality 3 + 2 = 5, as shown in the Table 3 - 4. Hence the result is true for n = 2. Note that, each of Ti [a, b, c], 1 ≤ i ≤ 9 such that (a, b, c) ∈ So (17, 3) has exactly 2+2 2 − 3 − (21+3 − 3) = 21+2 = 8 pendant vertices more than those in T0 [3, 3, 3]. Hence 6

V (T1 [3, 3, 11]) u, v, w ui , 1 ≤ i ≤ 2 vj , j = 1 wk , 1 ≤ k ≤ 10

V (T2 [3, 11, 3]) u, v, w ui , 1 ≤ i ≤ 2 vj , 1 ≤ j ≤ 9 wk , 1 ≤ k ≤ 2 V (T3 [3, 5, 9]) u, v, w ui , 1 ≤ i ≤ 2 vj , 1 ≤ j ≤ 3 wk , 1 ≤ k ≤ 8

V (T4 [5, 9, 3]) u, v, w ui , 1 ≤ i ≤ 4 vj , 1 ≤ j ≤ 7 wk , 1 ≤ k ≤ 2 V (T5 [9, 3, 5]) u, v, w ui , 1 ≤ i ≤ 8

vj , j = 1 wk , 1 ≤ k ≤ 4 V (T6 [3, 7, 7]) u, v, w ui , 1 ≤ i ≤ 2 vj , 1 ≤ j ≤ 5 wk , 1 ≤ k ≤ 6

Table 3:

f2 : V (T1 [3, 3, 11]) → P(X2 ) − {∅} f2 (V (T1 [3, 3, 11])) f2 (u) = f1 (u), f2 (v) = f1 (v), f2 (w) = f1 (w) f2 (u1 ) = f1 (u1 ), f2 (u2 ) = f1 (u2 ) f2 (v1 ) = f1 (v1 ) f2 (w1 ) = f1 (w1 ) , f2 (w2 ) = f1 (w2 ), f2 (w3 ) = {x5 } , f2 (w4 ) = {x1 , x5 }, f2 (w5 ) = {x2 , x5 } , f2 (w6 ) = {x4 , x5 }, f2 (w7 ) = {x1 , x2 , x5 } , f2 (w8 ) = {x1 , x4 , x5 }, f2 (w9 ) = {x2 , x4 , x5 } , f2 (w10 ) = {x1 , x2 , x4 , x5 }. f2 : V (T2 [3, 11, 3]) → P(X2 ) − {∅} f2 (V (T2 [3, 11, 3])) f2 (u) = f1 (u), f2 (v) = f1 (v), f2 (w) = f1 (w) f2 (u1 ) = f1 (u1 ), f2 (u2 ) = f1 (u2 ) f2 (v1 ) = f1 (v1 ), f2 (v2 ) = {x5 }, f2 (v3 ) = {x1 , x5 } , f2 (v4 ) = {x3 , x5 }, f2 (v5 ) = {x4 , x5 } , f2 (v6 ) = {x1 , x3 , x5 }, f2 (v7 ) = {x1 , x4 , x5 }, f2 (v8 ) = {x3 , x4 , x5 }, f2 (v9 ) = {x1 , x3 , x4 , x5 }. f2 (w1 ) = f1 (w1 ) , f2 (w2 ) = f1 (w2 ) f2 : V (T3 [3, 5, 9]) → P(X2 ) − {∅} f2 (V (T3 [3, 5, 9])) f2 (u) = f1 (u), f2 (v) = f1 (v), f2 (w) = f1 (w) f2 (u1 ) = f1 (u1 ), f2 (u2 ) = f1 (u2 ) f2 (v1 ) = f1 (v1 ), f2 (v2 ) = {x5 } , f2 (v3 ) = {x3 , x5 }, f2 (w1 ) = f1 (w1 ) , f2 (w2 ) = f1 (w2 ), f2 (w3 ) = {x1 , x5 } , f2 (w4 ) = {x4 , x5 }, f2 (w5 ) = {x1 , x2 , x5 } , f2 (w6 ) = {x1 , x4 , x5 }, f2 (w7 ) = {x2 , x4 , x5 }, f2 (w8 ) = {x1 , x2 , x4 , x5 }. f2 : V (T4 [5, 9, 3]) → P(X2 ) − {∅} f2 (V (T4 [5, 9, 3])) f2 (u) = f1 (u), f2 (v) = f1 (v), f2 (w) = f1 (w) f2 (u1 ) = f1 (u1 ), f2 (u2 ) = f1 (u2 ), f2 (u3 ) = {x5 } , f2 (u4 ) = {x2 , x5 } f2 (v1 ) = f1 (v1 ), f2 (v2 ) = {x3 , x5 } , f2 (v3 ) = {x4 , x5 }, f2 (v4 ) = {x1 , x3 , x5 }, f2 (v5 ) = {x1 , x4 , x5 }, f2 (v6 ) = {x3 , x4 , x5 }, f2 (v7 ) = {x1 , x3 , x4 , x5 }. f2 (w1 ) = f1 (w1 ) , f2 (w2 ) = f1 (w2 ) f2 : V (T5 [9, 3, 5]) → P(X2 ) − {∅} f2 (V (T5 [9, 3, 5])) f2 (u) = f1 (u), f2 (v) = f1 (v), f2 (w) = f1 (w) f2 (u1 ) = f1 (u1 ), f2 (u2 ) = f1 (u2 ), f2 (u3 ) = {x2 , x5 } , f2 (u4 ) = {x4 , x5 } f2 (u5 ) = {x2 , x3 , x5 } , f2 (u6 ) = {x2 , x4 , x5 }, f2 (u7 ) = {x3 , x4 , x5 } , f2 (u8 ) = {x2 , x3 , x4 , x5 } f2 (v1 ) = f1 (v1 ) f2 (w1 ) = f1 (w1 ) , f2 (w2 ) = f1 (w2 ), f2 (w3 ) = {x5 } , f2 (w4 ) = {x1 , x5 } f2 : V (T6 [3, 7, 7]) → P(X2 ) − {∅} f2 (V (T6 [3, 7, 7])) f2 (u) = f1 (u), f2 (v) = f1 (v), f2 (w) = f1 (w) f2 (u1 ) = f1 (u1 ), f2 (u2 ) = f1 (u2 ) f2 (v1 ) = f1 (v1 ), f2 (v2 ) = {x5 }, f2 (v3 ) = {x3 , x5 } , f2 (v4 ) = {x1 , x4 , x5 } f2 (v5 ) = {x1 , x3 , x4 , x5 }. f2 (w1 ) = f1 (w1 ) , f2 (w2 ) = f1 (w2 ), f2 (w3 ) = {x1 , x5 }, f2 (w4 ) = {x4 , x5 } f2 (w5 ) = {x1 , x2 , x5 } , f2 (w6 ) = {x2 , x4 , x5 }.

Set-sequential labeling of T1 [3, 3, 11], T2 [3, 11, 3], T3 [3, 5, 9], T4 [5, 9, 3], T5 [9, 3, 5], T6 [3, 7, 7].

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V (T7 [7, 3, 7]) u, v, w ui , 1 ≤ i ≤ 6 vj , 1 ≤ j ≤ 1 wk , 1 ≤ k ≤ 6

V (T8 [5, 5, 7]) u, v, w ui , 1 ≤ i ≤ 4 vj , 1 ≤ j ≤ 3 wk , 1 ≤ k ≤ 6

V (T9 [5, 7, 5]) u, v, w ui , 1 ≤ i ≤ 4 vj , 1 ≤ j ≤ 5 wk , 1 ≤ k ≤ 4

f2 : V (T7 [7, 3, 7]) → P(X2 ) − {∅} f2 (V (T7 [7, 3, 7])) f2 (u) = f1 (u), f2 (v) = f1 (v), f2 (w) = f1 (w) f2 (u1 ) = f1 (u1 ), f2 (u2 ) = f1 (u2 ), f2 (u3 ) = {x5 } , f2 (u4 ) = {x3 , x5 } f2 (u5 ) = {x2 , x4 , x5 } , f2 (u6 ) = {x2 , x3 , x4 , x5 } f2 (v1 ) = f1 (v1 ) f2 (w1 ) = f1 (w1 ) , f2 (w2 ) = f1 (w2 ), f2 (w3 ) = {x2 , x5 } , f2 (w4 ) = {x4 , x5 } f2 (w5 ) = {x1 , x2 , x5 } , f2 (w6 ) = {x1 , x4 , x5 }. f2 : V (T8 [5, 5, 7]) → P(X2 ) − {∅} f2 (V (T8 [5, 5, 7])) f2 (u) = f1 (u), f2 (v) = f1 (v), f2 (w) = f1 (w) f2 (u1 ) = f1 (u1 ), f2 (u2 ) = f1 (u2 ), f2 (u3 ) = {x5 } , f2 (u4 ) = {x3 , x5 } f2 (v1 ) = f1 (v1 ), f2 (v2 ) = {x4 , x5 } , f2 (v3 ) = {x3 , x4 , x5 } f2 (w1 ) = f1 (w1 ) , f2 (w2 ) = f1 (w2 ), f2 (w3 ) = {x2 , x5 } , f2 (w4 ) = {x1 , x2 , x5 } f2 (w5 ) = {x1 , x4 , x5 }, f2 (w6 ) = {x1 , x2 , x4 , x5 }. f2 : V (T9 [5, 7, 5]) → P(X2 ) − {∅} f2 (V (T9 [5, 7, 5])) f2 (u) = f1 (u), f2 (v) = f1 (v), f2 (w) = f1 (w) f2 (u1 ) = f1 (u1 ), f2 (u2 ) = f1 (u2 ), f2 (u3 ) = {x5 } , f2 (u4 ) = {x3 , x5 } f2 (v1 ) = f1 (v1 ), f2 (v2 ) = {x4 , x5 }, f2 (v3 ) = {x1 , x4 , x5 }, f2 (v4 ) = {x3 , x4 , x5 }, f2 (v5 ) = {x1 , x3 , x4 , x5 } f2 (w1 ) = f1 (w1 ), f2 (w2 ) = f1 (w2 ), f2 (w3 ) = {x2 , x5 } , f2 (w4 ) = {x1 , x2 , x5 }.

Table 4:

Set-sequential labeling of T7 [7, 3, 7], T8 [5, 5, 7], T9 [5, 7, 5].

any Ti [a, b, c], 1 ≤ i ≤ 9 such that (a, b, c) ∈ So (17, 3) can be obtained from T0 [3, 3, 3] by introducing 8 new pendant vertices and making them adjacent to the internal vertices of T0 [3, 3, 3] in a manner such that the degree of each internal vertex after such an addition is odd. Let the result be true for n > 1, that is, for (a, b, c) ∈ So (2n+2 + 1, 3) all the caterpillars T [a, b, c] of diameter four are set-sequential with respect to the set Xn = Xn−1 ∪ {xn+3 } = {x1 , x2 , . . . , xn+3 }, and |Xn | = n + 3. Clearly T [a, b, c] such that (a, b, c) ∈ So (2n+2 + 1, 3) is of the order 2n+2 and contains 2n+2 − 3 pendant vertices. We shall now prove that the result is true for n + 1 > 1. That is, if (a′ , b′ , c′ ) ∈ So (2n+3 + 1, 3) then, T [a′ , b′ , c′ ] is set-sequential with respect to Xn+1 = Xn ∪ {xn+4 }, where |Xn+1 | = n + 4. Let T [a′ , b′ , c′ ] any caterpillar such that (a′ , b′ , c′ ) ∈ So (2n+3 + 1, 3) and Xn+1 = Xn ∪ {xn+4 }, where |Xn+1 | = n + 4 be a non empty set. Clearly T [a′ , b′ , c′ ] has exactly 2n+3 − 3 − (2n+2 − 3) = 2n+2 pendant vertices more than T [a, b, c]. Therefore T [a′ , b′ , c′ ] can be obtained from T [a, b, c] by introducing 2n+2 isolated vertices and making them adjacent to any of the internal vertices u, v and w of T [a, b, c], which is set-sequential under the hypothesis, in any 3-even partition of 2n+2 , but in view of Proposition 2.2 it is enough to consider the partition of 2n+2 of the form {2n+2 , 0, 0} or {2n+1 , 2n+1 , 0} or {2n , 2n , 2n+1 }. Case 1. When T [a′ , b′ , c′ ] is obtained making 2n+2 isolated vertices adjacent to any one of the internal vertices u, v or w of set-sequentially labeled T [a, b, c]. Without loss of generality let it be adjacent to u, where fn (u) = fn−1 (u) = · · · = f1 (u) = {x1 }. T [a, b, c] being set-sequential with respect to Xn , to show that T [a′ , b′ , c′ ] which is obtained making 2n+2 isolated vertices adjacent to the internal vertex u of T [a, b, c] is also set-sequential

8

with respect to Xn+1 = Xn ∪ {xn+4 }, it is enough to show that the unlabeled 2n+2 vertices adjacent to u can be assigned the subsets of P(Xn+1 ) − P(Xn ) in an injective manner so that edges adjacent to them receives distinct elements of P(Xn+1 ) − P(Xn ) as the symmetric difference of its end vertices. Let Y = P(Xn+1 ) − P(Xn ), then |Y | = 2n+3 and there are exactly 2n+2 subsets of Xn+1 in Y which contains {x1 } and exactly 2n+2 subsets of Xn+1 in Y which does not contains {x1 }. Thus Y can be partitioned into two classes A and B such that A contains all the subsets of Xn+1 in Y which contains {x1 } and B contains all the subsets of Xn+1 in Y which do not contains {x1 }, hence |A| = |B| = 2n+2 . By assigning all the 2n+2 elements of B to the 2n+2 unlabeled pendant vertices adjacent to u in T [a′ , b′ , c′ ] in one-to-one manner, T [a′ , b′ , c′ ] is seen to be set-sequential. Similar argument follows for the case when T [a′ , b′ , c′ ] is obtained from set-sequentially labeled T [a, b, c] by making 2n+2 isolated vertices adjacent to either v or w in T [a, b, c]. Thus in each of the case T [a′ , b′ , c′ ] is seen to be set sequential. Case 2. When T [a′ , b′ , c′ ] is obtained making 2n+1 isolated vertices adjacent to any two of the internal vertices u, v, and w of set-sequentially labeled T [a, b, c]. Without loss of generality let T [a′ , b′ , c′ ] is obtained by making 2n+1 isolated vertices adjacent to u and v of T [a, b, c], where fn (u) = fn−1 (u) = · · · = f1 (u) = {x1 } and fn (v) = fn−1 (v) = · · · = f1 (v) = {x2 }. Note these 2n+1 new pendant vertices can be partitioned into 2n−1 copies each of which contains four vertices. Let Ci and Ci′ denote the partition of 2n+1 new pendant vertices adjacent to u and v each containing four vertices, where 1 ≤ i ≤ 2n−1 . To show that the unlabeled 2n+1 new pendant vertices adjacent to u and v can be assigned the subsets of P(Xn+1 ) − P(Xn ) in an injective manner so that edges adjacent to them receives distinct elements of P(Xn+1 ) − P(Xn ) as the symmetric difference of its end vertices, consider Yn+1 = Xn+1 − {x1 , x2 , x3 , x4 } = {x5 , x6 , . . . , xn+4 }, and Yn = Xn − {x1 , x2 , x3 , x4 } = {x5 , x6 , . . . , xn+3 }, thus |Yn+1 | = n and |Yn | = n − 1 which implies |P(Yn+1 )| = 2n and |P(Yn )| = 2n−1 . Let Y ′ = P(Yn+1 ) − P(Yn ), thus |Y ′ | = |P(Yn+1 ) − P(Yn )| = 2n−1 . Let Ai ∈ Y ′ ; 1 ≤ i ≤ 2n−1 . For each partition Ci of vertices  adjacent to u we make the following assignment: fn+1 (u1,i ) = {Ai }    fn+1 (u2,i ) = {x2 } ∪ {Ai } Ci : f (u ) = {x3 , x4 } ∪ {Ai }    n+1 3,i fn+1 (u4,i ) = {x2 , x3 , x4 } ∪ {Ai } and for each partition Ci′ of vertices adjacent to v we make the following assignment: fn+1 (v1,i ) = {x3 } ∪ {Ai }    fn+1 (v2,i ) = {x4 } ∪ {Ai } Ci′ : f (v ) = {x1 , x3 } ∪ {Ai }    n+1 3,i fn+1 (v4,i ) = {x1 , x4 } ∪ {Ai } where Ai ∈ Y ′ and 1 ≤ i ≤ 2n−1 . Clearly the assignment defined above is injective and is a set-sequential labeling of T [a′ , b′ , c′ ] with respect to Xn+1 , when T [a′ , b′ , c′ ] is obtained by making 2n+1 isolated vertices adjacent to u and v of set-sequentially labeled T [a, b, c]. For the case when T [a′ , b′ , c′ ] is obtained by making 2n+1 isolated vertices adjacent to the internal vertices u and w of set-sequentially labeled T [a, b, c], in the assignment described as above replacing x1 by x3 , x2 by x1 and x3 by x2 we get the set-sequential labeling of T [a′ , b′ , c′ ]. Finally, when T [a′ , b′ , c′ ] is obtained by making 2n+1 isolated vertices adjacent to the internal vertices v and w of set-sequentially labeled T [a, b, c], in the assignment described as above replacing 9

x1 by x2 , x2 by x3 and x3 by x1 we get the set-sequential labeling of T [a′ , b′ , c′ ]. Thus in either of the cases T [a′ , b′ , c′ ] is seen to be set-sequential. Case 3. When T [a′ , b′ , c′ ] is obtained making 2n isolated vertices adjacent to any two of the internal vertices u, v, and w and 2n+1 isolated vertices adjacent to any one of the remaining internal vertices u, v, and w of set-sequentially labeled T [a, b, c]. Without loss of generality let T [a′ , b′ , c′ ] is obtained by making 2n isolated vertices adjacent to v and w of T [a, b, c] and 2n+1 isolated vertices adjacent to u, where fn (u) = fn−1 (u) = · · · = f1 (u) = {x1 }, fn (v) = fn−1 = · · · = f1 (v) = {x2 } and fn (w) = fn−1 (w) = · · · = f1 (w) = {x3 }. Since, 2n+1 new pendant vertices can be partitioned into 2n−1 copies each of which contains four vertices and 2n new pendant vertices can be partitioned into 2n−1 copies each of which contains two vertices. Let Ci denote the partition of unlabeled 2n+1 pendant vertices adjacent to u each containing four vertices, where 1 ≤ i ≤ 2n−1 and Cj and Cj′ denote the partition of unlabeled 2n pendant vertices adjacent to v and w each containing two vertices, where 1 ≤ j ≤ 2n−1 . For each partition Ci of vertices adjacent to u we make the  following assignment: fn+1 (u1,i ) = {x3 } ∪ {Ai }    fn+1 (u2,i ) = {x2 , x3 } ∪ {Ai } Ci : f (u ) = {x2 , x4 } ∪ {Ai }    n+1 3,i fn+1 (u4,i ) = {x2 , x3 , x4 } ∪ {Ai }, where Ai ∈ Y ′ and 1 ≤ i ≤ 2n−1 . For each partition Cj of vertices adjacent to v we make the { following assignment: fn+1 (v1,j ) = {Aj } Cj : fn+1 (v2,j ) = {x1 } ∪ {Aj } where Aj ∈ Y ′ and 1 ≤ j ≤ 2n−1 , and for each partition Cj′ of vertices adjacent to w we make { the following assignment: fn+1 (w1,j ) = {x4 } ∪ {Aj } Cj′ : fn+1 (w2,j ) = {x1 , x4 } ∪ {Aj } where Aj ∈ Y ′ and 1 ≤ j ≤ 2n−1 . Clearly the assignment defined above is injective and is a set-sequential labeling of T [a′ , b′ , c′ ] with respect to Xn+1 , when T [a′ , b′ , c′ ] is obtained by making 2n isolated vertices adjacent to v and w and 2n+1 isolated vertices adjacent to u of set-sequentially labeled T [a, b, c]. For the case when T [a′ , b′ , c′ ] is obtained by making 2n isolated vertices adjacent to the internal vertices u and w and 2n+1 isolated vertices adjacent to v of of set-sequentially labeled T [a, b, c], in the assignment described as above replacing x1 by x2 , x2 by x3 and x3 by x1 we get the set-sequential labeling of T [a′ , b′ , c′ ]. Finally, when T [a′ , b′ , c′ ] is obtained by making 2n isolated vertices adjacent to the internal vertices u and v, and 2n+1 isolated vertices adjacent to w of of set-sequentially labeled T [a, b, c], in the assignment described as above replacing x1 by x3 , x2 by x1 and x3 by x2 we get the set-sequential labeling of T [a′ , b′ , c′ ]. Thus in either of the cases T [a′ , b′ , c′ ] is seen to be set-sequential. Corollary 2.1. for some n ≥ 1.

T [a, b, c] + K1 is set-graceful if and only if (a, b, c) ∈ So (2n+2 + 1, 3)

We now introduce a class of tree of diameter four. By a star lobster we would mean a tree with the property that the removal of its endpoints leaves a star K1,n , denoted as L[d1 , d2 , . . . , dn ], where d1 , d2 , . . . , dn are the degrees of the vertices u1 , u2 , . . . , un which 10

are not the end vertices or central vertex. It is easy to see that star lobster is of diameter 4. Set-sequential L[d1 , d2 , . . . , dn ] for di = 3 : 1 ≤ i ≤ n is characterized as Theorem 2.3. [8] L[d1 , d2 , . . . , dm ] for di = 3; 1 ≤ i ≤ m is set-sequential if and only if 3m ≡ 0(mod 2n − 1). Using arguments similar to proposition 2.2 it is verifiable that the following is true Proposition 2.3.

So (2n+4 − 1, 3) is generated by So (2n+3 − 1, 3) for n ≥ 1.

Our next result characterizes set-sequential L[a, b, c]. Theorem 2.4. some n ≥ 1.

L[a, b, c] is set-sequential if and only if (a, b, c) ∈ So (2n+3 − 1, 3) for

Proof. The condition is necessary in view of Acharya-Hegde necessary condition and Theorem 1.3. Conversely, let L[a, b, c] be such that (a, b, c) ∈ So (2n+3 − 1, 3) for some n ≥ 1. Let the central vertex of L[a, b, c] be denoted as z and the u, v, w be the vertices of which are not the pendant vertices, and pendant vertices adjacent to u, v, and w respectively be denoted as ui where 1 ≤ i ≤ a − 1, vj where 1 ≤ j ≤ b − 1 and wk where 1 ≤ k ≤ c − 1. Since, d(u) + d(v) + d(w) + d(z) = a + b + c + 3 = 2n+3 + 2 it is easily seen that L[a, b, c] is of the order 2n+3 , and the number of pendant vertices are 2n+3 − 4. We shall show that L[a, b, c] is set-sequential. For n = 1, all the non-isomorphic tress associated with So (15, 3) are shown in the Table 5 and each of them has 21+3 − 4 = 12 pendant vertices. Non isomorphic trees L[a, b, c] such that (a, b, c) ∈ So (15, 3) L11 [3, 3, 9], L12 [3, 9, 3] 1 L3 [3, 5, 7], L14 [3, 7, 5], L15 [5, 3, 7] L16 [5, 5, 5]

Table 5:

All Non isomorphic trees L[a, b, c] such that (a, b, c) ∈ So (15, 3).

They are seen to be set-sequential with respect to X1 = {x1 , x2 , x3 , x4 } ∪ {x5 }, |X1 | = 4 + 1 = 5 as shown in Table 6. For n = 2, all the non-isomorphic tress associated with So (31, 3) and are shown in the Table 7 and each of them has 22+3 − 4 = 28 pendant vertices. Thus each of L2i [a, b, c], 1 ≤ i ≤ 31 where (a, b, c) ∈ So (31, 3) has exactly 22+3 −4−(21+3 −4) = 22+2 = 16 pendant vertices more than any L1j [a′ , b′ , c′ ], 1 ≤ j ≤ 6 where (a′ , b′ , c′ ) ∈ So (15, 3). Note that except for L223 [5, 9, 17], L224 [9, 17, 5], L225 [17, 5, 9] each of L2i [a, b, c] such that (a, b, c) ∈ So (31, 3) can be obtained from L1j ; 1 ≤ j ≤ 6 by making adjacent 16 new pendant vertices to u, v, w in L1j , by considering any partition of 21+3 = 16 of the form {16, 0, 0} or {8, 8, 0} or {4, 4, 8}. Also the vertices u, v, w in L1j [a, b, c] are labeled as f1 (u) = {x1 }, f1 (v) = {x2 } and f1 (w) = {x3 } which is same as the labels of the internal vertices of T [a, b, c] described in Theorem 2.2. So using the assignment scheme described in Case 1 - 3 of Theorem 2.2, each of L2i other than L223 [5, 9, 17], L224 [9, 17, 5], L225 [17, 5, 9] is seen to be set-sequential. For L223 [5, 9, 17], L224 [9, 17, 5], L225 [17, 5, 9] set-sequential labeling is as shown in Table 8. 11

(L11 [3, 3, 9])

V z, u, v, w ui , 1 ≤ i ≤ 2 vj , 1 ≤ j ≤ 2 wk , 1 ≤ k ≤ 8

V (L12 [3, 9, 3]) z, u, v, w ui , 1 ≤ i ≤ 2 vj , 1 ≤ j ≤ 8 wk , 1 ≤ k ≤ 2 V (L13 [3, 5, 7]) z, u, v, w ui , 1 ≤ i ≤ 2 vj , 1 ≤ j ≤ 4 wk , 1 ≤ k ≤ 6 V (L14 [3, 7, 5]) z, u, v, w ui , 1 ≤ i ≤ 2 vj , 1 ≤ j ≤ 6 wk , 1 ≤ k ≤ 4 V (L15 [5, 3, 7]) z, u, v, w ui , 1 ≤ i ≤ 4 vj , 1 ≤ j ≤ 2 wk , 1 ≤ k ≤ 6 V (L16 [5, 5, 5]) z, u, v, w ui , 1 ≤ i ≤ 4 vj , 1 ≤ j ≤ 4 wk , 1 ≤ k ≤ 4

Table 6:

→ P(X1 ) − {∅} f1 (V f1 (z) = {x1 , x2 , x3 }, f1 (u) = {x1 }, f1 (v) = {x2 }, f1 (w) = {x3 } f1 (u1 ) = {x4 }, f1 (u2 ) = {x3 , x4 } f1 (v1 ) = {x5 }, f1 (v2 ) = {x3 , x5 } f1 (w1 ) = {x1 , x5 }, f1 (w2 ) = {x4 , x5 }, f1 (w3 ) = {x1 , x2 , x4 } f1 (w4 ) = {x1 , x2 , x5 }, f1 (w5 ) = {x1 , x4 , x5 }, f1 (w6 ) = {x2 , x3 , x4 } f1 (w7 ) = {x2 , x4 , x5 }, f1 (w8 ) = {x1 , x2 , x4 , x5 }. f1 : V (L12 [3, 9, 3]) → P(X1 ) − {∅} 1 f1 (V (L2 [3, 9, 3])) f1 (z) = {x1 , x2 , x3 }, f1 (u) = {x1 }, f1 (v) = {x2 }, f1 (w) = {x3 } f1 (u1 ) = {x4 }, f1 (u2 ) = {x3 , x4 } f1 (v1 ) = {x5 }, f1 (v2 ) = {x1 , x5 }, f1 (v3 ) = {x3 , x5 }, f1 (v4 ) = {x4 , x5 }, f1 (v5 ) = {x1 , x3 , x5 }, f1 (v6 ) = {x1 , x4 , x5 }, f1 (v7 ) = {x3 , x4 , x5 }, f1 (v8 ) = {x1 , x3 , x4 , x5 }. f1 (w1 ) = {x1 , x2 , x4 }, f1 (w2 ) = {x2 , x3 , x4 } f1 : V (L13 [3, 5, 7]) → P(X1 ) − {∅} 1 f1 (V (L3 [3, 5, 7])) f1 (z) = {x1 , x2 , x3 }, f1 (u) = {x1 }, f1 (v) = {x2 }, f1 (w) = {x3 } f1 (u1 ) = {x4 }, f1 (u2 ) = {x3 , x4 } f1 (v1 ) = {x5 }, f1 (v2 ) = {x3 , x5 }, f1 (v3 ) = {x1 , x4 , x5 }, f1 (v4 ) = {x1 , x3 , x4 , x5 }, f1 (w1 ) = {x1 , x5 }, f1 (w2 ) = {x4 , x5 }, f1 (w3 ) = {x1 , x2 , x4 }, f1 (w4 ) = {x1 , x2 , x5 }, f1 (w5 ) = {x2 , x3 , x4 }, f1 (w6 ) = {x2 , x4 , x5 }. f1 : V (L14 [3, 7, 5]) → P(X1 ) − {∅} 1 f1 (V (L4 [3, 7, 5])) f1 (z) = {x1 , x2 , x3 }, f1 (u) = {x1 }, f1 (v) = {x2 }, f1 (w) = {x3 } f1 (u1 ) = {x4 }, f1 (u2 ) = {x3 , x4 } f1 (v1 ) = {x1 , x5 }, f1 (v2 ) = {x4 , x5 }, f1 (v3 ) = {x1 , x3 , x5 } f1 (v4 ) = {x1 , x4 , x5 }, f1 (v5 ) = {x3 , x4 , x5 }, f1 (v6 ) = {x1 , x3 , x4 , x5 } f1 (w1 ) = {x5 }, f1 (w2 ) = {x2 , x5 }, f1 (w3 ) = {x1 , x2 , x4 }, f1 (w4 ) = {x2 , x3 , x4 }. f1 : V (L15 [5, 3, 7]) → P(X1 ) − {∅} 1 f1 (V (L5 [5, 3, 7])) f1 (z) = {x1 , x2 , x3 }, f1 (u) = {x1 }, f1 (v) = {x2 }, f1 (w) = {x3 } f1 (u1 ) = {x4 }, f1 (u2 ) = {x5 }, f1 (u3 ) = {x3 , x4 }, f1 (u4 ) = {x3 , x5 }, f1 (v1 ) = {x4 , x5 }, f1 (v2 ) = {x3 , x4 , x5 }, f1 (w1 ) = {x2 , x5 }, f1 (w2 ) = {x1 , x2 , x4 }, f1 (w3 ) = {x1 , x2 , x5 } f1 (w4 ) = {x1 , x4 , x5 }, f1 (w5 ) = {x2 , x3 , x4 }, f1 (w6 ) = {x1 , x2 , x4 , x5 }. f1 : V (L16 [5, 5, 5]) → P(X1 ) − {∅} 1 f1 (V (L6 [5, 5, 5])) f1 (z) = {x1 , x2 , x3 }, f1 (u) = {x1 }, f1 (v) = {x2 }, f1 (w) = {x3 } f1 (u1 ) = {x4 }, f1 (u2 ) = {x5 }, f1 (u3 ) = {x3 , x4 }, f1 (u4 ) = {x3 , x5 }, f1 (v1 ) = {x4 , x5 }, f1 (v2 ) = {x1 , x4 , x5 }, f1 (v3 ) = {x3 , x4 , x5 }, f1 (v4 ) = {x1 , x3 , x4 , x5 }, f1 (w1 ) = {x2 , x5 }, f1 (w2 ) = {x1 , x2 , x4 }, f1 (w3 ) = {x1 , x2 , x5 } f1 (w4 ) = {x2 , x3 , x4 }. f1 : V (L11 [3, 3, 9]) 1 (L1 [3, 3, 9]))

Set-sequential labeling of L11 [3, 3, 9], L12 [3, 9, 3], L13 [3, 5, 7], L14 [3, 7, 5], L15 [5, 3, 7], L16 [5, 5, 5].

12

Non isomorphic trees L[a, b, c] such that (a, b, c) ∈ So (31, 3) L21 [3, 3, 25], L22 [3, 25, 3] 2 L3 [3, 5, 23], L24 [5, 23, 3], L25 [23, 3, 5] L26 [3, 7, 21], L27 [7, 21, 3], L28 [21, 3, 7] L29 [3, 9, 19], L210 [9, 19, 3], L211 [19, 3, 9] 2 L12 [3, 11, 17], L213 [11, 17, 3], L214 [17, 3, 11] L215 [3, 13, 15], L216 [13, 15, 3], L217 [15, 3, 13] L218 [5, 5, 21], L219 [5, 21, 5] 2 L20 [5, 7, 19], L221 [7, 19, 5], L222 [19, 5, 7] L223 [5, 9, 17], L224 [9, 17, 5], L225 [17, 5, 9] 2 L26 [5, 11, 15], L227 [11, 15, 5], L228 [15, 5, 11] L229 [5, 13, 13], L230 [13, 5, 13] L231 [7, 7, 17], L232 [7, 17, 7] 2 L33 [7, 9, 15], L234 [9, 15, 7], L235 [15, 7, 9] 2 L36 [7, 11, 13], L237 [11, 13, 7], L238 [13, 7, 11] L239 [9, 9, 13], L240 [9, 13, 9] L241 [9, 11, 11], L242 [11, 9, 11]

Table 7:

All the non isomorphic trees L[a, b, c] associated with (a, b, c) ∈ So (31, 3).

Thus all the trees L[a, b, c] for (a, b, c) ∈ S0 (31, 3) are seen to be set-sequential for the set X2 = X1 ∪ {x6 } = {x1 , x2 , x3 , x4 } ∪ {x5 } ∪ {x6 }, |X2 | = 4 + 2 = 6. Let the result be true for n > 1, that is, for (a, b, c) ∈ So (2n+3 − 1, 3) all the star lobsters L[a, b, c] are set-sequential with respect to the set Xn = Xn−1 ∪ {xn+4 }, where |Xn | = 4 + n = n + 4. Clearly L[a, b, c] is of the order 2n+3 and contains 2n+3 − 4 pendant vertices. We shall now prove that the result is true for n + 1 > 1. That is, if (a1 , b1 , c1 ) ∈ So (2n+4 − 1, 3) then, L[a1 , b1 , c1 ] is set-sequential with respect to Xn+1 = Xn ∪ {xn+5 }, where |Xn+1 | = 4 + (n + 1) = n + 5. Let L[a1 , b1 , c1 ] be such that (a1 , b1 , c1 ) ∈ So (2n+4 − 1, 3) and Xn+1 = Xn ∪ {xn+5 }, where |Xn+1 | = n + 5 be any set. Clearly L[a1 , b1 , c1 ] has exactly 2n+4 − 4 − (2n+3 − 4) = 2n+3 pendant vertices more than any of L[a, b, c] such that (a, b, c) ∈ So (2n+3 − 1, 3). Therefore L[a1 , b1 , c1 ] can be obtained from L[a, b, c] by introducing 2n+3 isolated vertices and making them adjacent to any of the vertices u, v and w of L[a, b, c], which is set-sequential under the hypothesis, in any 3-even partition of 2n+3 , but in view of Proposition 2.3 it is enough to consider the partition of 2n+3 of the form {2n+3 , 0, 0} or {2n+2 , 2n+2 , 0} {2n+1 , 2n+1 , 2n+2 }. Case 1. When L[a1 , b1 , c1 ] is obtained by making 2n+3 isolated vertices adjacent to any one of the vertex u, v, w of set-sequentially labeled L[a, b, c]. With argument similar as in case (1) of Theorem 2.2, L[a1 , b1 , c1 ] is seen to be set-sequential. Case 2. When L[a1 , b1 , c1 ] is obtained by making 2n+2 isolated vertices adjacent to any two of the vertices u, v, w of set-sequentially labeled L[a, b, c]. Without loss of generality let L[a1 , b1 , c1 ] is obtained by making 2n+2 isolated vertices adjacent to u and v of L[a, b, c], where fn (u) = fn−1 (u) = · · · = f1 (u) = {x1 }, and fn (v) = fn−1 (v) = · · · = f1 (v) = {x2 }. Note these 2n+2 new pendant vertices can be partitioned into 2n copies each of which contains four vertices. Let Ci and Ci′ denote the partition of 2n+2 new 13

(L223 [5, 9, 17])

V z, u, v, w ui , 1 ≤ i ≤ 4 vj , 1 ≤ j ≤ 8

wk , 1 ≤ k ≤ 16

V (L224 [9, 17, 5]) z, u, v, w ui , 1 ≤ i ≤ 8 vj , 1 ≤ j ≤ 16

wk , 1 ≤ k ≤ 4 V (L225 [17, 5, 9]) z, u, v, w ui , 1 ≤ i ≤ 16

vj , 1 ≤ j ≤ 4 wk , 1 ≤ k ≤ 8

→ P(X2 ) − {∅} f2 (V f2 (z) = {x1 , x2 , x3 }, f2 (u) = {x1 }, f2 (v) = {x2 }, f2 (w) = {x3 } f2 (u1 ) = {x4 }, f2 (u2 ) = {x5 }, f2 (u3 ) = {x3 , x4 }, f2 (u4 ) = {x3 , x5 }, f2 (v1 ) = {x6 }, f2 (v2 ) = {x4 , x5 }, f2 (v3 ) = {x3 , x6 }, f2 (v4 ) = {x5 , x6 } f2 (v5 ) = {x1 , x4 , x5 }, f2 (v6 ) = {x3 , x4 , x5 }, f2 (v7 ) = {x3 , x5 , x6 } f2 (v8 = {x1 , x3 , x4 , x5 }, f2 (w1 ) = {x1 , x6 }, f2 (w2 ) = {x2 , x5 }, f2 (w3 ) = {x4 , x6 }, f2 (w4 ) = {x1 , x2 , x4 }, f2 (w5 ) = {x1 , x2 , x5 }, f2 (w6 ) = {x1 , x2 , x6 }, f2 (w7 ) = {x1 , x4 , x6 }, f2 (w8 ) = {x1 , x5 , x6 }, f2 (w9 ) = {x2 , x3 , x4 }, f2 (w10 ) = {x2 , x4 , x6 }, f2 (w11 ) = {x4 , x5 , x6 }, f2 (w12 ) = {x1 , x2 , x4 , x6 }, f2 (w13 ) = {x1 , x2 , x5 , x6 }, f2 (w14 ) = {x1 , x4 , x5 , x6 }, f2 (w15 ) = {x2 , x4 , x5 , x6 }, f2 (w16 ) = {x1 , x2 , x4 , x5 , x6 }. f1 : V (L224 [9, 17, 5]) → P(X2 ) − {∅} 2 f2 (V (L24 [9, 17, 5])) f2 (z) = {x1 , x2 , x3 }, f2 (u) = {x1 }, f2 (v) = {x2 }, f2 (w) = {x3 } f2 (u1 ) = {x4 }, f2 (u2 ) = {x6 }, f2 (u3 ) = {x1 , x5 }, f2 (u4 ) = {x2 , x6 }, f2 (u5 ) = {x3 , x4 }, f2 (u6 ) = {x3 , x5 }, f2 (u7 ) = {x5 , x6 }, f2 (u8 ) = {x2 , x5 , x6 }, f2 (v1 ) = {x3 , x6 }, f2 (v2 ) = {x4 , x5 }, f2 (v3 ) = {x4 , x6 }, f2 (v4 ) = {x1 , x3 , x6 }, f2 (v5 ) = {x1 , x4 , x5 }, f2 (v6 ) = {x1 , x4 , x6 }, f2 (v7 ) = {x3 , x4 , x5 }, f2 (v8 ) = {x3 , x4 , x6 }, f2 (v9 ) = {x3 , x5 , x6 }, f2 (v10 ) = {x4 , x5 , x6 }, f2 (v11 ) = {x1 , x3 , x4 , x5 }, f2 (v12 ) = {x1 , x3 , x4 , x6 }, f2 (v13 ) = {x1 , x3 , x5 , x6 }, f2 (v14 ) = {x1 , x4 , x5 , x6 }, f2 (v15 ) = {x3 , x4 , x5 , x6 }, f2 (v16 ) = {x1 , x3 , x4 , x5 , x6 }, f2 (w1 ) = {x2 , x5 }, f2 (w2 ) = {x1 , x2 , x4 }, f2 (w3 ) = {x1 , x2 , x5 } f2 (w4 ) = {x2 , x3 , x4 }. f1 : V (L225 [17, 5, 9]) → P(X2 ) − {∅} f2 (V (L225 [17, 5, 9])) f2 (z) = {x1 , x2 , x3 }, f2 (u) = {x1 }, f2 (v) = {x2 }, f2 (w) = {x3 } f2 (u1 ) = {x4 }, f2 (u2 ) = {x1 , x5 }, f2 (u3 ) = {x2 , x6 }, f2 (u4 ) = {x3 , x4 }, f2 (u5 ) = {x3 , x5 }, f2 (u6 ) = {x4 , x6 }, f2 (u7 ) = {x2 , x3 , x6 }, f2 (u8 ) = {x2 , x4 , x6 }, f2 (u9 ) = {x2 , x5 , x6 }, f2 (u10 ) = {x3 , x4 , x6 }, f2 (u11 ) = {x4 , x5 , x6 }, f2 (u12 ) = {x2 , x3 , x4 , x6 }, f2 (u13 ) = {x2 , x3 , x5 , x6 }, f2 (u14 ) = {x2 , x4 , x5 , x6 }, f2 (u15 ) = {x3 , x4 , x5 , x6 }, f2 (u16 ) = {x2 , x3 , x4 , x5 , x6 }, f2 (v1 ) = {x4 , x5 }, f2 (v2 ) = {x1 , x4 , x5 }, f2 (v3 ) = {x3 , x4 , x5 }, f2 (v4 ) = {x1 , x3 , x4 , x5 }, f2 (w1 ) = {x5 }, f2 (w2 ) = {x1 , x6 }, f2 (w3 ) = {x2 , x5 }, f2 (w4 ) = {x5 , x6 }, f2 (w5 ) = {x1 , x2 , x4 }, f2 (w6 ) = {x1 , x2 , x5 }, f2 (w7 ) = {x1 , x5 , x6 }, f2 (w8 ) = {x2 , x3 , x4 }, f2 : V (L223 [5, 9, 17]) 2 (L23 [5, 9, 17]))

Table 8:

Set-sequential labeling of L223 [5, 9, 17], L224 [9, 17, 5], L225 [17, 5, 9].

pendant vertices adjacent to u and v each containing four vertices, where 1 ≤ i ≤ 2n . To show that the unlabeled 2n+2 new pendant vertices adjacent to u and v can be assigned the subsets of P(Xn+1 ) − P(Xn ) in an injective manner so that edges adjacent to them receives distinct elements of P(Xn+1 ) − P(Xn ) as the symmetric difference of its end vertices, consider Yn+1 = Xn+1 − {x1 , x2 , x3 , x4 } = {x5 , x6 , . . . , xn+5 }, and Yn = Xn − {x1 , x2 , x3 , x4 } = {x5 , x6 , . . . , xn+4 }, thus |Yn+1 | = n + 1 and |Yn | = n which implies that |P(Yn+1 )| = 2n+1 and |P(Yn )| = 2n . Let Y1 = P(Yn+1 ) − P(Yn ), thus 14

|Y1 | = |P(Yn+1 ) − P(Yn )| = 2n . Let Ai ∈ Y1 ; 1 ≤ i ≤ 2n . For each each partition Ci of vertices  adjacent to u we make the following assignment: fn+1 (u1,i ) = {Ai }    fn+1 (u2,i ) = {x2 } ∪ {Ai } Ci : f (u ) = {x3 , x4 } ∪ {Ai }    n+1 3,i fn+1 (u4,i ) = {x2 , x3 , x4 } ∪ {Ai } and for each partition Ci′ of vertices adjacent to v we make the following assignment: fn+1 (v1,i ) = {x3 } ∪ {Ai }    f n+1 (v2,i ) = {x4 } ∪ {Ai } Ci′ : f (v ) = {x1 , x3 } ∪ {Ai }    n+1 3,i fn+1 (v4,i ) = {x1 , x4 } ∪ {Ai } where Ai ∈ Y ′ and 1 ≤ i ≤ 2n . Clearly the assignment defined above is injective and is a set-sequential labeling of L[a1 , b1 , c1 ] with respect to Xn+1 , when L[a1 , b1 , c1 ] is obtained by making 2n+2 vertices adjacent to u and v of set-sequentially labeled L[a, b, c]. For the case when L[a1 , b1 , c1 ] obtained by making 2n+2 isolated vertices adjacent to the internal vertices u and w of set-sequentially labeled L[a, b, c], in the assignment described as above replacing x1 by x3 , x2 by x1 and x3 by x2 we get the set-sequential labeling of L[a1 , b1 , c1 ]. Finally, when L[a1 , b1 , c1 ] is obtained by making 2n+2 isolated vertices adjacent to the internal vertices v and w of setsequentially labeled L[a, b, c], in the assignment described as x1 by x2 , x2 by x3 and x3 by x1 we get the set-sequential labeling of L[a1 , b1 , c1 ]. Thus in either of the cases L[a1 , b1 , c1 ] is seen to be set-sequential. Case 3. When L[a1 , b1 , c1 ] is obtained making 2n+1 isolated vertices adjacent to any two of the vertices u, v, and w and 2n+2 isolated vertices adjacent to any one of the remaining vertices u, v, and w of set-sequentially labeled L[a, b, c]. Without loss of generality let L[a1 , b1 , c1 ] obtained by making 2n+1 isolated vertices adjacent to v and w and 2n+2 isolated vertices adjacent to u of set-sequentially labeled L[a, b, c], where fn (u) = fn−1 (u) = · · · = f1 (u) = {x1 }, fn (v) = fn−1 (v) = · · · = f1 (v) = {x2 } and fn (w) = fn−1 (w) = · · · = f1 (w) = {x3 }. Since, 2n+2 new pendant vertices can be partitioned into 2n copies each of which contains four vertices and 2n+1 new vertices can be partitioned into 2n copies each of which contains two vertices. Let Ci denote the partition of unlabeled 2n+2 new vertices adjacent to u each containing four vertices, where 1 ≤ i ≤ 2n and Cj and Cj′ denote the partition of unlabeled 2n+1 new vertices adjacent to v and w each containing two vertices where 1 ≤ j ≤ 2n . For each partition Ci of vertices adjacent to u we make the following assignment:  f (u1,i ) = {x3 } ∪ {Ai }  n+1   fn+1 (u2,i ) = {x2 , x3 } ∪ {Ai } Ci : f (u ) = {x2 , x4 } ∪ {Ai }    n+1 3,i fn+1 (u4,i ) = {x2 , x3 , x4 } ∪ {Ai }, where Ai ∈ Y ′ and 1 ≤ i ≤ 2n . For each partition Cj of vertices adjacent to v we make the { following assignment: fn+1 (v1,j ) = {Aj } Cj : fn+1 (v2,j ) = {x1 } ∪ {Aj } where Aj ∈ Y ′ and 1 ≤ j ≤ 2n , and for each partition Cj′ of vertices adjacent to w we make the following assignment:

15

{ fn+1 (w1,j ) = {x4 } ∪ {Aj } : fn+1 (w2,j ) = {x1 , x4 } ∪ {Aj } where Aj ∈ Y ′ and 1 ≤ j ≤ 2n . Clearly the assignment defined above is injective and is a set-sequential labeling of L[a1 , b1 , c1 ] with respect to Xn+1 , when L[a1 , b1 , c1 ] is obtained by making 2n+1 isolated vertices adjacent to v and w and 2n+2 isolated vertices adjacent to u of set-sequentially labeled L[a, b, c]. For the case when L[a1 , b1 , c1 ] is obtained by making 2n+1 isolated vertices adjacent to the vertices u and w and 2n+2 isolated vertices adjacent to vertex v of set-sequentially labeled L[a, b, c], in the assignment described as above replacing x1 by x2 , x2 by x3 and x3 by x1 we get the set-sequential labeling of L[a1 , b1 , c1 ]. Finally, when L[a1 , b1 , c1 ] is obtained by making 2n+1 isolated vertices adjacent to the vertices u and v and 2n+2 isolated vertices adjacent to w of set-sequentially labeled L[a, b, c], in the assignment described as above replacing x1 by x3 , x2 by x1 and x3 by x2 we get the set-sequential labeling of L[a1 , b1 , c1 ]. Thus in either of the cases L[a1 , b1 , c1 ] is be set-sequential. Hence the proof follows. Cj′

As an immediate consequence of the foregoing two theorems and Theorem 1.1 we have the following Corollary 2.2. L[d1 , d2 , . . . , dn ] + K1 for di = 3; 1 ≤ i ≤ n is set-graceful if and only if 3m ≡ 0(mod 2n − 1). Corollary 2.3. for some n ≥ 1.

L[a, b, c] + K1 is set-graceful if and only if (a, b, c) ∈ So (2n+3 − 1, 3)

Note that in Case (1) of the proof of Theorem 2.2, if 2n+2 pendant vertices are made adjacent to the end vertex w1 of the diametral path of set-sequentially labeled T [a, b, c] with respect to a set Xn = Xn−1 ∪ {xn } = {x1 , x2 , . . . , xn }, and |Xn | = n + 3, such that f (w1 ) = {x4 }, then we get a set-sequential caterpillar T [a, b, c, d] of diameter five with respect to a set Xn+1 = Xn ∪ {xn+1 } = {x1 , x2 , . . . , xn+1 }, and |Xn | = n + 4 and one of the end vertex of a diametral path in T [a, b, c, d] is necessarily assigned with {xn+1 }. Thus if T is a set-sequentially labeled caterpillar of diameter d, then by adding 2m pendant vertices to the end vertex of the diametral path in T, where 2m = |V (T )| we get a setsequential caterpillar of diameter d + 1. Also set-sequential trees up to diameter 3 has been completely characterized [8]. Thus following is immediate. Theorem 2.5.

For every positive integer d there is a set-sequential tree of diameter d.

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[4] B. D. Acharya, On d-sequential graphs, J. Math. and Phys. Sci., (1983) 17(1), 21-35. [5] B. D. Acharya, Characterization of eulerian set-sequential graphs, submmited. [6] B. D. Acharya and S.M. Hegde, Set-sequential graphs, Nat. Acad. Sci. Letters, (1985) 8(12), 387-390. [7] B. D. Acharya, K. A. Germina, K. L. Princy and S. B. Rao, On set-valuations of graphs, Labeling of Discrete Structures and Applications, Eds. B.D. Acharya, S. Arumugan & Alexender Rosa, Narosa Pub. House, New Delhi, 2008. [8] B. D. Acharya, Germina K. A, Kumar Abhishek and P. J. Slater, Some new results on set-graceful and set-sequential graphs, Journal of Combinatorics, Information & System Sciences (JCISS), Accepted. [9] F. Harary, Graph Theory, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1969. [10] Germina K. A. and Kumar Abhishek, On Set-Sequential Caterpillars, subbmited. [11] Hansraj Gupta, Slected Topics in Number Theory, Abascus Press, England, 1979. [12] M. Mollard and C. Payan, On two conjectures about set-graceful graphs, European J. Combin. (1989) 10, 185-187. [13] P. J. Slater, On k-sequential and other numbered graphs, Discrete Math. (1981) 34, 185-193. http://dx.doi.org/10.1016/0012-365X(81)90066-2 [14] S. M. Hegde, On set-valuations of graphs, Nat. Acad. Sci. Letters, (1991) 14 (4), 181-182. [15] S. M. Hegde, Set colorings of graphs., Eur. J. Comb., (2009) 30 (4), 986-995. http://dx.doi.org/10.1016/j.ejc.2008.06.005 [16] P. N. Balister, E. Gyri, and R. H. Schelp, Coloring vertices and edges of a graph by nonempty subsets of a set, Eur. J. Comb., (2011) 32 (4), 533-537. http://dx.doi.org/10.1016/j.ejc.2010.11.008

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