Math.Comput.Sci. (2013) 7:237–243 DOI 10.1007/s11786-013-0151-z
Mathematics in Computer Science
Face Antimagic Labeling of Jahangir Graph Muhammad Kamran Siddiqui · Muhammad Numan · Muhammad Awais Umar
Received: 27 February 2013 / Accepted: 20 March 2013 / Published online: 13 April 2013 © Springer Basel 2013
Abstract This paper deals with the problem of labeling the vertices, edges and faces of a plane graph. A weight of a face is the sum of the label of a face and the labels of the vertices and edges surrounding that face. In a super d-antimagic labeling the vertices receive the smallest labels and the weights of all s-sided faces constitute an arithmetic progression of difference d, for each s that appearing in the graph. The paper examines the existence of super d-antimagic labelings for Jahangir graphs for certain differences d. Keywords
Plane graph · d-antimagic labeling · Jahangir graph
Mathematics Subject Classification (2010)
Primary 05C78; Secondary 05C38
1 Introduction and Definitions Let G = (V, E, F) be a finite connected plane graph without loops and multiple edges, where V (G), E(G) and F(G) are its vertex set, edge set and face set, respectively. General reference for graph-theoretic notions is [17]. A labeling of type (1, 1, 1) assigns labels from the set {1, 2, . . . , |V (G)| + |E(G)| + |F(G)|} to the vertices, edges and faces of plane graph G in such a way that each vertex, edge and face receives exactly one label and each number is used exactly once as a label. The weight of a face under a labeling of type (1, 1, 1) is the sum of labels carried by that face and all the edges and vertices surrounding it.
The work was supported by Higher Education Commission Pakistan Grant HEC(FD)/2012/555. M. K. Siddiqui (B) · M. Numan · M. A. Umar Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan e-mail:
[email protected] M. Numan e-mail:
[email protected] M. A. Umar e-mail:
[email protected]
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A labeling of type (1, 1, 1) of a plane graph G is called d-antimagic if for every positive integer s the set of weights of all s-sided faces is Ws = {as , as + d, as + 2d, . . . , as + ( f s − 1)d} for some integers as and d ≥ 0, where f s is the number of the s-sided faces. We allow different sets Ws for different s. Somewhat related types of antimagic labelings were defined by Hartsfield and Ringel in [11] and by Bodendiek and Walther in [10]. The concept of the d-antimagic labeling of plane graphs was defined in [8]. This labeling is a natural extension of the notion of magic labeling (i.e. 0-antimagic labeling) defined by Lih [12]. The d-antimagic labelings of type (1, 1, 1) for the hexagonal planar maps, generalized Petersen graph P(n, 2) and grids can be found in [4,6] and [7], respectively. Lin et al. [13] and Sugeng et al. [15] described d-antimagic labelings of type (1, 1, 1) for prisms Dn . A d-antimagic labeling is called super if the smallest possible labels appear on the vertices. The super d-antimagic labelings of type (1, 1, 1) for antiprisms and for disjoint union of prisms are presented in [2] and [3]. The existence of super d-antimagic labeling of type (1, 1, 1) for the plane graphs containing a special Hamilton path is examined in [5] and super d-antimagic labelings of type (1, 1, 1) for disconnected plane graphs are investigated in [9]. In the paper we examine the existence of super d-antimagic labelings for Jahangir graphs. The Jahangir graph Jn,m , n ≥ 3, m ≥ 1, consists of a cycle Cnm and one additional vertex which is adjacent to n vertices of Cnm at distance m to each other on Cnm . The Jahangir graph was introduced by Tomescu and Javaid [16], where is studied its metric dimension. A total vertex irregularity strength of Jn,m can be found in [1]. The Jahangir graph Jn,2 is also known as the gear graph, see Ma and Feng [14]. For m = 1, the Jahangir graph is wheel Wn .
2 Antimagic Labelings of Jahangir Graph The Jahangir graph Jn,m , n ≥ 3, m ≥ 1, has n vertices of degree 3, n(m − 1) vertices of degree 2, one vertex of degree n, n(m + 1) edges, n internal (m + 2)-sided faces and one external mn-side face. Let the vertex set, edge set and face set of Jn,m be defined as follows (see Fig. 1): V (Jn,m ) = c, vi , u ij : 1 ≤ i ≤ n, 1 ≤ j ≤ m − 1 , E(Jn,m ) = cvi , vi u i1 : 1 ≤ i ≤ n ∪ u im−1 vi+1 : 1 ≤ i ≤ n − 1 ∪ n u m−1 v1 ∪ u ij u ij+1 ; 1 ≤ i ≤ n, 1 ≤ j ≤ m − 2 , F(Jn,m ) = { f 1 , f 2 , . . . , f n , f ext }. Note that under the study of d-antimagic labeling of Jn,m we will consider a weight of the external mn-side face only if m = 1 and n = 3. In other cases the external mn-side face is unified and comparing its weight to weights of internal (m + 2)-sided faces is without sense. The first theorem shows the existence of a super d-antimagic labeling of type (1, 1, 1) for wheel Wn = Jn,1 . Theorem 1 For d ∈ {0, 1, 2, 3, 4} and for every n ≥ 3, the graph Wn = Jn,1 has a super d-antimagic labeling of type (1, 1, 1). Proof First we define labelings ψd of type (1, 1, 1) which assign labels from the set {1, 2, . . . , 4n + 2} to the vertices, edges and faces of the graph Wn = Jn,1 . For every d ∈ {0, 1, 2, 3, 4} we put ψd (c) = 1, ψd (vi ) = 1 + i, for 1 ≤ i ≤ n, ψd (vi vi+1 ) = 2 + 2n − i, for 1 ≤ i ≤ n − 1, ψd (vn v1 ) = n + 2, ψd ( f ext ) = 4n + 2.
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Fig. 1 The graph of Jn,m
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The spokes and internal 3-sided faces are labeled as follows. 3n + 1 − i, if 1 ≤ i ≤ n − 1 ψ0 (cvi ) = 3n + 1, if i = n 3n + 2 + i, if 1 ≤ i ≤ n − 1 ψ0 ( f i ) = 3n + 2, if i = n 4n − 2i, if 1 ≤ i ≤ n − 1 ψ1 (cvi ) = 4n, if i = n 2n + 3 + 2i, if 1 ≤ i ≤ n − 1 ψ1 ( f i ) = 2n + 3, if i = n ψ2 (cvi ) = 2n + 1 + i, ψ3 (cvi ) = 2n + 2i, if 1 ≤ i ≤ n 3n + 2, if i = 1 ψ2 ( f i ) = 4n + 3 − i, if 2 ≤ i ≤ n 2n + 3, if i = 1 ψ3 ( f i ) = 4n + 5 − 2i, if 2 ≤ i ≤ n 3n + 1 + i, if 1 ≤ i ≤ n − 1 ψ4 (cvi ) = 3n + 1, if i = n 2n + 1 + i, if 1 ≤ i ≤ n − 1 ψ4 ( f i ) = 4n + 1, if i = n For d ∈ {0, 1, 2, 3, 4} let us denote wtd ( f i+1 ) = ψd (c) + ψd (vi ) + ψd (vi+1 ) + ψd (vi vi+1 ) + ψd (cvi ) + ψd (cvi+1 ) + ψd ( f i+1 ) as a weight of the internal 3-sided face f i+1 , 1 ≤ i ≤ n − 1, and wtd ( f 1 ) = ψd (c) + ψd (vn ) + ψd (v1 ) + ψd (vn v1 ) + ψd (cvn ) + ψd (cv1 ) + ψd ( f 1 ) as a weight of the internal 3-sided face f 1 and
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as a weight of the external n-sided face. Case d = 0. Under the labeling ψ0 , all the weights of the internal 3-sided faces have the same value wt0 ( f i ) = 11n + 10, 1 ≤ i ≤ n and wt0 ( f ext ) = 2n 2 + 7n + 2. However, if n = 3 then the weight of external 3-sided face is not 11n + 10. If we modify the face labels such that ψ0 ( f 1 ) = 13, ψ0 ( f 2 ) = 14, ψ0 ( f 3 ) = 12, ψ0 ( f ext ) = 11 and swap the edge value ψ0 (v2 v3 ) = 6 with the face value ψ0 ( f 3 ) = 12, then all the 3-sided faces in the graph W3 have the same weight 44. Thus we obtain a super 0-antimagic labeling of type (1, 1, 1) for Wn for all n ≥ 3. Cases d = 1 and d = 2. In the labeling ψ1 (respectively, ψ2 ), the weights for the 3-sided faces constitute an arithmetic progression 11n + 10, 11n + 11, . . . , 12n + 9 (respectively, 10n + 11, 10n + 13, . . . , 12n + 9) and the weight of the external n-sided face is again 2n 2 +7n +2. For n = 3, wt1 ( f ext ) = wt2 ( f ext ) = 41 and in both cases it can not be a member of the arithmetic progression with difference d = 1 (respectively, d = 2) of the weights of 3-sided faces. For d = 1, if we modify the face labels such that ψ1 ( f 1 ) = 9, ψ1 ( f 3 ) = 11 and swap the edge value ψ1 (v2 v3 ) = 6 with the face value ψ1 ( f 3 ) = 11, then the weight of the face f ext will increase by 5 and the weights of all 3-sided faces constitute the arithmetic progression 43, 44, 45, 46. For d = 2, if we swap the edge value ψ2 (v1 v2 ) = 7, with the face value ψ2 ( f 2 ) = 13, then the weights of 3-sided faces form the arithmetic sequence with difference 2. It follows that we have a super 1-antimagic (respectively, super 2-antimagic) labeling of type (1, 1, 1) for Wn for every n ≥ 3. Cases d = 3 and d = 4. If we use the labeling ψ3 (respectively, ψ4 ), then the weights of the 3-sided faces constitute an arithmetic progression with difference d = 3 (respectively, d = 4), namely 10n + 11, 10n + 14, . . . , 13n + 8 (respectively, 10n + 11, 10n + 15, . . . , 14n + 7), and wt3 ( f ext ) = wt4 ( f ext ) = 2n 2 + 7n + 2. For n = 3, the weights of the external 3-sided faces can not be any members of the arithmetic progression with difference d = 3 (respectively, d = 4). Swapping the edge label ψ3 (v2 v3 ) = 6 with the face value ψ3 ( f 3 ) = 11 and the edge label ψ3 (v3 v1 ) = 5 with the face value ψ3 ( f 1 ) = 9 (respectively, the each edge label ψ4 (vi vi+1 ) with the face value ψ4 ( f i+1 ) for i = 1, 2, 3), will increase the weights of the external 3-sided faces to 50 (respectively, 53) and then the all 3-sided faces form an arithmetic progression with difference 3 (respectively, 4). Next theorem shows the existence of a super d-antimagic labeling of type (1, 1, 1) for graph Jn,m , m ≥ 2. Theorem 2 The Jahangir graph Jn,m , m ≥ 2 and n ≥ 3, admits a super d-antimagic labeling of type (1, 1, 1) for d ∈ {0, 1, 2, 3, 4}. Proof Define a labeling ϕd : V (Jn,m ) ∪ E(Jn,m ) ∪ F(Jn,m ) → {1, 2, 3, . . . , 2mn + 2n + 2} for every d ∈ {0, 1, 2, 3, 4} in the following way: ϕd (vi ) = 1 + i, ϕd vi u i1 = n(m + 1) + 2 − i, for 1 ≤ i ≤ n, ϕd (c) = 1, ϕd u ij = 1 + i + n j, for 1 ≤ i ≤ n, 1 ≤ j ≤ m − 1, ϕd u ij u ij+1 = n(m + 1 + j) + 2 − i, for 1 ≤ i ≤ n, 1 ≤ j ≤ m − 2, ϕd u im−1 vi+1 = 2mn + 2 − i, for 1 ≤ i ≤ n − 1,
ϕd u nm−1 v1 = n(2m − 1) + 2, ϕd ( f ext ) = 2n(m + 1) + 2, and for spokes and internal (m + 2)-sided faces as follows: n(2m + 1) + 1 − i, if 1 ≤ i ≤ n − 1 ϕ0 (cvi ) = n(2m + 1) + 1, if i = n
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ϕ1 (cvi ) = ϕ1 ( f i ) = ϕ2 (cvi ) = ϕ2 ( f i ) = ϕ3 (cvi ) = ϕ3 ( f i ) = ϕ4 (cvi ) = ϕ4 ( f i ) =
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⎧ if i = 1 ⎪ ⎨ n(2m + 1) + 3, n(2m + 1) + 2 + i, if 2 ≤ i ≤ n − 1 ⎪ ⎩ n(2m + 1) + 2, if i = n 2n(m + 1) − 2i, if 1 ≤ i ≤ n − 1 2n(m + 1), if i = n 2mn + 3 + 2i, if 1 ≤ i ≤ n − 1 2mn + 3, if i = n 2mn + 1 + i, if 1 ≤ i ≤ n n(2m + 1) + 2, if i = 1 2n(m + 1) + 3 − i, if 2 ≤ i ≤ n 2mn + 2i, if 1 ≤ i ≤ n, 2mn + 3, if i = 1 2n(m + 1) + 5 − 2i, if 2 ≤ i ≤ n n(2m + 1) + 1 + i, if 1 ≤ i ≤ n − 1 n(2m + 1) + 1, if i = n 2mn + 1 + i, if 1 ≤ i ≤ n − 1 2n(m + 1) + 1, if i = n.
It is easy to verify that the weights of the internal (m + 2)-sided faces
wtd ( f 1 ) = ϕd (c) + ϕd (cvn ) + ϕd (vn ) + ϕd vn u n1 +
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+ϕd u nm−1 v1 + ϕd (v1 ) + ϕd (cv1 ), and wtd ( f i+1 ) = ϕd (c) + ϕd (cvi ) + ϕd (vi ) + ϕd vi u i1 +
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+ϕd u im−1 vi+1 + ϕd (vi+1 ) + ϕd (cvi+1 ), for 1 ≤ i ≤ n − 1, form an arithmetic sequence with the desired differences, more precisely for 1 ≤ i ≤ n: wt0 ( f i ) = n(2m 2 + 6m + 3) + 3m + 7, wt1 ( f i ) = n(2m 2 + 6m + 4) + 3m + 7 − i, wt2 ( f i ) = n(2m 2 + 6m + 2) + 3m + 6 + 2i, wt3 ( f i ) = n(2m 2 + 6m + 2) + 3m + 5 + 3i, wt4 ( f i ) = n(2m 2 + 6m + 2) + 3m + 4 + 4i. This completes the proof.
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Next we consider the labeling ϕd : V (Jn,m )∪ E(Jn,m )∪ F(Jn,m ) → {1, 2, 3, . . . , 2mn +2n +2}, for d ∈ {5, 6}, where ϕ5 (c) = ϕ6 (c) = 1, ϕ5 (vi ) = 2i and ϕ6 (vi ) = 3i − 1, for 1 ≤ i ≤ n, ϕ5 vi u i1 = 2mn + 3 − 2i and ϕ6 (vi u i1 ) = 2mn + 4 − 3i, for 1 ≤ i ≤ n, ϕ5 u im−1 vi+1 = ϕ6 (u im−1 vi+1 ) = n(m + 1) + 2 − i, for 1 ≤ i ≤ n − 1, and
ϕ5 u nm−1 v1 = ϕ6 (u nm−1 v1 ) = mn + 2, 2i + 1, if j = 1, 1 ≤ i ≤ n ϕ5 u ij = 1 + i + n j, if 2 ≤ j ≤ m − 1, 1 ≤ i ≤ n 3i + j − 1, if 1 ≤ j ≤ 2, 1 ≤ i ≤ n ϕ6 u ij = 1 + i + n j, if 3 ≤ j ≤ m − 1, 1 ≤ i ≤ n 2mn + 2 − 2i, if j = 1, 1 ≤ i ≤ n ϕ5 u ij u ij+1 = 2mn + 2 − i − jn, if 2 ≤ j ≤ m − 2, 1 ≤ i ≤ n 2mn + 4 − j − 3i, if 1 ≤ j ≤ 2, 1 ≤ i ≤ n i i ϕ6 u j u j+1 = 2mn + 2 − i − jn, if 3 ≤ j ≤ m − 2, 1 ≤ i ≤ n n(2m + 1) + 1 + i, if 1 ≤ i ≤ n − 1 ϕ5 (cvi ) = ϕ6 (cvi ) = n(2m + 1) + 1, if i = n 2mn + 1 + i, if 1 ≤ i ≤ n − 1 ϕ5 ( f i ) = ϕ6 ( f i ) = 2n(m + 1) + 1, if i = n ϕ5 ( f ext ) = ϕ6 ( f ext ) = 2n(m + 1) + 2. We can observe that the labeling ϕd , d ∈ {5, 6}, uses each integer from the given set exactly once and the smallest possible labels from the interval [1, mn + 1] appear on the vertices. Theorem 3 For m ≥ 3 and n ≥ 3, the graph Jn,m admits a super 5-antimagic labeling of type (1, 1, 1). Proof Label the vertices, edges and faces of Jn,m by the labeling ϕ5 . It is not difficult to verify that weights of the internal (m + 2)-sided faces form an arithmetic sequence n(2m 2 + 6m + 2) + 3m + 3 + 5i, 1 ≤ i ≤ n, and therefore the desired result follows. Theorem 4 For m ≥ 4 and n ≥ 3, the graph Jn,m admits a super 6-antimagic labeling of type (1, 1, 1). Proof We label the vertices, edges and faces of the graph Jn,m by the labeling ϕ6 . By direct computation we obtain that the weights of the internal (m + 2)-sided faces constitute the set {wt6 ( f i ) = n(2m 2 + 6m + 2) + 3m + 2 + 6i : 1 ≤ i ≤ n}. This proves that ϕ6 is a super 6-antimagic labeling. In the next theorem we show the existence of d-antimagic labelings of Jahangir graph for all d ≥ 7 but with restriction for m. Theorem 5 If m + 2 ≥ d ≥ 7 and n ≥ 3 then the graph Jn,m has a super d-antimagic labeling of type (1, 1, 1). Proof Let n ≥ 3 and m + 2 ≥ d ≥ 7. For vertices, edges and faces of Jn,m we define the labeling ϕ7 such that ϕ7 (vi ) = 2 + (d − 3)(i − 1) and ϕ7 vi u i1 = 2mn − 2 + d − (d − 3)i, for 1 ≤ i ≤ n,
ϕ7 u im−1 vi+1 = n(m + 1) + 2 − i, for 1 ≤ i ≤ n − 1 and ϕ7 u nm−1 v1 = mn + 2, 2 + (d − 3)(i − 1) + j, if 1 ≤ j ≤ d − 4, 1 ≤ i ≤ n ϕ7 u ij = 1 + i + n j, if d − 3 ≤ j ≤ m − 1, 1 ≤ i ≤ n 2mn + 1 − j − (d − 3)(i − 1), if 1 ≤ j ≤ d − 4, 1 ≤ i ≤ n ϕ7 u ij u ij+1 = 2mn + 2 − i − jn, if d − 3 ≤ j ≤ m − 2, 1 ≤ i ≤ n ϕ7 (cvi ) = ϕ5 (cvi ) and ϕ7 ( f i ) = ϕ5 ( f i ), ϕ7 (c) = ϕ5 (c) and ϕ7 ( f ext ) = ϕ5 ( f ext ).
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We can see that the labeling ϕ7 is a bijective function. The weights of the internal (m + 2)-sided faces constitute the set {wt7 ( f i ) = n(2m 2 + 6m + 2) + 3m + 8 + d(i − 1) : 1 ≤ i ≤ n} which implies that the labeling ϕ7 is a super d-antimagic labeling of type (1, 1, 1).
3 Concluding Remarks In the foregoing section we studied super d-antimagic labelings of type (1, 1, 1) for the Jahangir graph Jn,m and we showed the existence of such labelings for every d ≥ 0, but with restrictions. We are not able to prove or disprove the existence of such labelings for m + 1 ≤ d if d ≥ 7. So, we conclude the paper with the following open problem. Open Problem 1 For the graph Jn,m , 3 ≤ m + 1 ≤ d, n ≥ 3, determine if there is a super d-antimagic labeling of type (1, 1, 1) with d ≥ 7.
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