Multilevel Bandwidth and Radio Labelings of Graphs Riadh Khennoufa and Olivier Togni LE2I, UMR CNRS 5158 Universit´e de Bourgogne, 21078 Dijon cedex, France
[email protected],
[email protected]
Abstract. This paper introduces a generalization of the graph bandwidth parameter: for a graph G and an integer k ≤ diam(G), the k-level bandwidth B k (G) of G is defined by B k (G) = minγ max{|γ(x)−γ(y)|−d(x, y)+1 : x, y ∈ V (G), d(x, y) ≤ k}, the minimum being taken among all proper numberings γ of the vertices of G. We present general bounds on B k (G) along with more specific results for k = 2 and the exact value for k = diam(G). We also exhibit relations between the k-level bandwidth and radio k-labelings of graphs from which we derive a upper bound for the radio number of an arbitrary graph.
Keywords: generalized graph bandwidth, radio labeling, frequency assignment.
1
Introduction
Let G be a connected graph. The distance between two vertices u and v of G is denoted by dG (u, v) or simply d(u, v) and the diameter of G is denoted by diam(G). The graph bandwidth problem is an old and well studied NP-complete problem (see e.g. [10, 11, 4, 9, 5]) that remains NP-complete for several simple graphs like some special types of trees. This problem arises from sparse matrix computation, coding theory, and circuit layout of VLSI design. The bandwidth B(G) of a graph G is the minimum of the quantity max{|γ(x) − γ(y)| : xy ∈ E(G)} taken over all proper numberings γ of G. In this paper, we introduce a generalisation of the bandwidth parameter called k-level bandwidth which consists, for a proper numbering of the vertices, in taking into acount not only the differences of the labels of adjacent vertices, but also of vertices at distance i, 1 ≤ i ≤ k: for a graph G and an integer k ≤ diam(G), the k-level bandwidth B k (G, γ) of a proper numbering of the vertices of a graph G is defined by B k (G, γ) = max{|γ(x) − γ(y)| − d(x, y) + 1 : x, y ∈ V (G), d(x, y) ≤ k}. The k-level bandwidth B k (G) of G is B k (G) = min{B k (G, γ) : γ is a proper numbering of V (G)}.
A numbering γ for which B k (G, γ) = B k (G) is said to be a k-level bandwidth numbering. We shall also use the notation B ∗ (G) = B diam(G) (G). Thus, the 1-level bandwidth corresponds with the graph bandwidth: B 1 (G) = B(G) while the 2-level bandwidth B 2 generalises the bandwidth in an analogous manner as the λ2,1 parameter for the chromatic number. Figure 1 presents three numberings of a graph G: the left one γ1 is such that B(G, γ1 ) = 4, B 2 (G, γ1 ) = 5, the one on the center γ2 is such that B(G, γ2 ) = 4, B 2 (G, γ2 ) = 6 and the right one γ3 is such that B(G, γ3 ) = 3, B 2 (G, γ3 ) = 5.
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Fig. 1. Example of numberings of a graph with different bandwidths and 2-level bandwidths
Other labelings of interest since few years are radio labelings: for a graph G and an integer k, a radio k-labeling f of G is an assignment of non negative integers to the vertices of G such that |f (u) − f (v)| ≥ k + 1 − d(u, v), for every two distinct vertices u and v of G. The span of the function f denoted by λk (f ), is max{f (x) − f (y) : x, y ∈ V (G)}. The radio k-chromatic number λk (G) of G is the minimum span of all radio k-labelings of G. Radio k-labelings were introduced by Chartrand et al. [1], motivated by radio channel assignment problems with interference constraints. Quite few results are known concerning radio k-labelings. The radio k-chromatic number for paths was studied in [1], where lower and upper bounds were given. These bounds have been improved in [6]. Radio k-labelings generalize many other graph labelings. A radio 1-labeling is a proper vertex-colouring and λ1 (G) = χ(G) − 1. For k = 2, the radio 2labeling problem corresponds to the well studied L(2, 1)-labeling problem. For k = diam(G) − 1, radio k-labelings were studied under the name of (radio) antipodal labelings [1, 3, 7, 8]. A radio k-labeling with k = diam(G) is known as a radio labeling. Radio labelings of paths and of cycles have been studied in [2, 12]. The aim of this paper is to introduce the k-level bandwidth parameter, to raise links between it and the radio k-labeling problem and to present bounds on the k-level bandwidth of some graphs.
2
Multilevel bandwidth
To begin, we claim that for paths, cycles and complete bipartite graphs, the k-level bandwidth is easy to determine: Claim 1 – For any k ≤ n − 1, B k (Pn ) = 1; – for any k < n2 , B k (Cn ) = k + 1 and B p (C2p ) = p; – B 2 (Km,n ) = m + n − 2. If Gk stands for the k th power of G (i.e. the graph with the same vertex set as G and with edges between vertices at distance at most k in G), then the following relation is easily seen: Proposition 1 For any graph G and any k ≤ diam(G), B(G) ≤ B k (G) ≤ B(Gk ). Proposition 2 For any graph G of order n and any k, 1 ≤ k ≤ diam(G), B k (G) ≥
k(n − 1) − k + 1. diam(G)
Proof. Let γ : V (G) −→ {1, . . . , n} be a k-level bandwidth numbering of G and let u = γ −1 (1) and v = γ −1 (n). If d(u, v) ≤ k, then we have B k (G, γ) ≥ n − 1 − d(u, v) + 1 ≥ n − k ≥ k(n−1) diam(G) − k + 1 since k ≤ diam(G). If d(u, v) > k, then by the pigeonhole principle, there exist two vertices x and y at mutual distance k along a shortest path between u and v such that k(n−1) |γ(x) − γ(y)| ≥ diam(G) − k + 1. t u The Cartesian product G2H of graphs G and H is the graph with vertex set V (G) × V (H) and edge set E(G2H) = {((a, x)(b, y)), ab ∈ E(G) and x = y or xy ∈ E(H) and a = b}. The following theorem generalize the known upper bound (see [5]) on the bandwidth of the Cartesian product of two graphs: Theorem 1 For any graphs G and H, and for any positive integer k ≥ 1, B k (G2H) ≤ |V (G)|B k (H)+max{(|V (G)|−1)(k−1), B k (G)+(|V (G)|−1)(k−2)−1}. Proof. The numbering of G2H which gives this upper bound is γ(a, x) = γG (a)+ (γH (x)−1)|V (G)|, where γG and γH are k-level bandwidth numberings of G and of H, respectively. Due to space constraints, the rest of the proof is left to the reader. Corollary 1 For any graph G of diameter at least 2, B 2 (G2K2 ) ≤ 2B 2 (G) + 1.
. Theorem 2 For any graph G of order n and any k ≤ diam(G), B k (G) ≤ n − diam(G). Proof. Let u and v be vertices of G such that d(u, v) = diam(G) and let u0 = u, u1 , u2 , . . . , un−1 = v be a distance ordering of V (G) from u0 , i.e. if i < j then d(u0 , ui ) ≤ d(u0 , uj ). Let also Vi = {uj ∈ V (G), d(u0 , uj ) = i} and ni = |Vi |, 0 ≤ i ≤ diam(G). Thus, the Vi partition V (G) into diam(G) + 1 levels, each level Vi represent the set of vertices which are at distance i with the vertex u0 . Now, we show that the simple numbering γ of the vertices of G given by γ(ui ) = i + 1 attains the desired bandwidth, i.e. that the the condition |γ(y) − γ(x)| − d(x, y) + 1 ≤ n − diam(G) is verified for any two vertices x and y, with d(x, y) ≤ k. Assume w.l.o.g. that x ∈ Vi (G) and y ∈ Vj (G) for some i and j, 0 ≤ i ≤ j ≤ diam(G) and j − i ≤ k. Then we have Pi−1 γ(x) ≥ 1 + `=0 n` , and Pj γ(y) ≤ `=0 n` . Thus, γ(y) − γ(x) − d(x, y) + 1 ≤ n − diam(G) ⇔ Pi−1 n − 1 − `=0 n` − d(x, y) + 1 ≤ n − diam(G) ⇔ ` `=0 Pj `=i n` − d(x, y) ≤ n − diam(G).
Pj
Pj Pj As d(x, y) ≥ j − i, we have `=i n` − d(x, y) ≤ `=i (n` − 1) + 1 ≤ Pdiam(G) Pdiam(G) Pdiam(G) (n − 1) + 1 = n + (1) − 1 = n − diam(G). Hence ` ` `=0 `=0 `=0 the above inequality is verified and we obtain B k (G) ≤ n − diam(G). t u Combining Theorem 2 and Proposition 2 with k = diam(G), we obtain the exact value of B ∗ (G) i.e. of B diam(G) (G): Corollary 2 For any graph G of order n, B ∗ (G) = n − diam(G).
3
Relation between the radio k-chromatic number and the k-level bandwidth
Theorem 3 For any graph G of order n and for any positive integer k ≥ 1, λk (G) ≤ λk+B
k
(G)−1
(Pn ).
Proof. Let γ be a k-level bandwidth numbering of G and let f : V (Pn ) → 0 {0, . . . , λk (Pn ) − 1} be a radio k 0 -labeling of Pn , with k 0 = k + B k (G) − 1. Consider the vertex ordering u1 , u2 , . . . , un of G induced by γ: ui = γ −1 (i) and label each vertex ui using f , as if u1 , . . . , un was a path. Then, we have for i > j, |f (ui ) − f (uj )| ≥ k 0 + 1 − dPn (ui , uj ) = k 0 + 1 − (i − j) = k + B k (G) − (i − j). As B k (G) ≥ |γ(ui ) − γ(uj )| − dG (ui , uj ) + 1 = (i − j) − dG (ui , uj ) + 1, we obtain |f (ui ) − f (uj )| ≥ k + (i − j) − dG (ui , uj ) + 1 − (i − j) = k + 1 − dG (ui , uj ), and thus f is a radio k-labeling of G.
t u
This result, along with the upper bound of Chartrand et al. for the radio k-chromatic number of the path [1] yield the following: Corollary 3 For any graph G of order n and for any positive integer 1 ≤ k ≤ diam(G), 1 λk (G) ≤ (B k (G)(B k (G) + 2k) + k 2 − 1). 2 Corollary 4 For any graph G of order n and diameter D, 1 2 (n − 2n + 2) if n is even, λD (G) ≤ 21 2 2 (n − 2n + 5) if n is odd. Proof. By using the result given in Theorem 2 and applying it in Theorem 3 ∗ with k = diam(G) = D, we obtain λD (G) ≤λD+B (G)−1 (Pn ) = λn−1 (Pn ). Liu 1 (n2 − 2n + 2) if n is even, and Zhu [12], have shown that λn−1 (Pn ) = 12 2 . u t 2 (n − 2n + 5) if n is odd.
4
The 2-level bandwidth B 2
For the 2-level bandwidth, one can derive a better lower bound than that of Proposition 2 in some cases by refining the argument of the proof: Proposition 3 For any graph G of diameter at least 2, n−1 1 B 2 (G) ≥ 2 − − 1. diam(G) 2 Since it is easily seen that B 2 (C4 = H2 ) = 2, Corollary 1 gives a upper bound for the 2-level bandwidth of the hypercube: Proposition 4 For the n-dimensional hypercube Hn , n ≥ 2, B 2 (Hn ) ≤ 2n−1 + n − 2.
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Concluding remarks
We have introduced the k-level bandwidth of a graph and presented first results about it. Nevertheless, many questions and problems remain open. Among all: – We have shown that the k-level bandwidth is easy to determine when k = diam(G) and it is known that computing the 1-level bandwidth is NPcomplete. An open problem is thus to determine the algorithmic complexity of computing B k (G) for 2 ≤ k ≤ diam(G) − 1. – Another interesting question is: given a graph G, does there exist a numbering γ of V (G) which is optimal for the k-level bandwidth (i.e. such that B k (G, γ) = B k (G)) for any k ?
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