Discrete Applied Mathematics

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1 Discrete Applied Matheatics 7 (009) Contents lists available at ScienceDirect Discrete Applied Matheatics journal hoepage: Equitable total coloring of C C n Tong Chunling ab Lin Xiaohui a Yang Yuansheng a Li Zhihe a a Departent of Coputer Science and Engineering Dalian University of Technology Dalian 60 PR China b Departent of Inforation Science and Engineering Shandong Jiaotong University Jinan 00 PR China a r t i c l e i n f o a b s t r a c t Article history: Received 6 May 007 Received in revised for 8 August 008 Accepted August 008 Available online Septeber 008 The equitable total chroatic nuber of a graph G is the sallest integer k for which G has a k-total coloring such that the nuber of vertices and edges colored with each color differs by at ost one. In this paper we show that the Cartesian product graphs of C and C n have equitable total -coloring for all and n. 008 Elsevier B.V. All rights reserved. Keywords: Total coloring Equitable total coloring Equitable total chroatic nuber Cycle Cartesian product. Introduction We consider only finite undirected graphs without loops or ultiple edges. Let G = (V(G) E(G)) be a graph with vertex set V and edge set E. A k-total coloring of a graph G is a coloring of the vertices and edges of G with k colors so that no two adjacent or incident eleents have the sae color. The total chroatic nuber χ (G) of G is the sallest k such that G has a k-total coloring. A total coloring is equitable if the nuber of vertices and edges colored with each color differs by at ost one. The sallest k for which G has such a coloring is naed equitable total chroatic nuber and denoted by χ = (G). There is a long -standing Total Coloring Conjecture (TCC) forulated by Behzad [] and Vizing [] independently which says that (G) + χ (G) (G) + for a siple graph G. Sánchez-Arroyo [] has shown that deciding χ (G) is NP-coplete. McDiarid and Sánchez-Arroyo [0] have shown that deterining the total chroatic nuber of k-regular bipartite graphs is NP-hard for each fixed k too. TCC has been verified for several classes of graphs in recent years [697]. Kostochka [8] proved that the total coloring of any ultigraph with axial degree is 6-total colorable. Fu [] first investigated the equitable total coloring. He raised Conjecture. and proved it for a few special cases such as trees coplete graphs coplete bipartite graphs coplete split graphs and graphs G with (G) V(G). Conjecture.. For any graph G G has an equitable total k-coloring for each k ax(χ (G) (G) + ). Wang [6] put forward Conjecture. and proved it for all ultigraphs G with (G). Conjecture.. For any graph G χ = (G) (G) +. Research supported by NFSC (6070). Corresponding author. Tel.: E-ail address: yangys@dlut.edu.cn (Y. Yang) X/$ see front atter 008 Elsevier B.V. All rights reserved. doi:0.06/j.da

2 C. Tong et al. / Discrete Applied Matheatics 7 (009) Fig... C C. Zhang [9] obtained the equitable total chroatic nuber of soe join graphs. The Cartesian product G H of two graphs G and H is the graph with vertex set V(G) V(H) in which the vertex (a b) is adjacent to the vertex (c d) whenever a = c and b is adjacent to d or b = d and a is adjacent to c. Let C and C n be cycles of length and n respectively the Cartesian product C C n is a -regular graph with n vertices v ij 0 i n and 0 j where indices i and j are read odulo n and respectively. Fig.. shows C C. A lot of work has been done on the total chroatic nuber of Cartesian product graphs [78]. Seoud et al. [] proved that χ (C C n ) = + for and n being an even nuber or a ultiple of. Kenitz and Marangio [7] further proved that χ (C C n ) = + holds for and n being a ultiple of too. Baril et al. [] proved that any Cartesian product of cycles has an adjacent vertex distinguishing chroatic index equal to + i.e. that the graph is type by the observation ade in [] and also in [0]. In this paper we will show that χ = (C C n ) = + = for all and n.. Equitable total -coloring of C C n Now we consider the coloring of C C n. We have Theore.. C C n has equitable total -coloring for all and n. Proof. By syetry we need only consider the cases for od n od. Let V = {v ij : 0 i n 0 j } E = {v ij v ij+ : 0 i n 0 j } E n = {v ij v i+j : 0 i n 0 j } where indices i and j are read odulo n and respectively. Let β = β = β = β = β =. Let σ be a coloring of C C n as follows: Case. 0 od. Sv = β β β β β Se = β β β β β Se n = β β β β β. Case.. n 0 od (see Fig..()). σ (V) = (Sv ) n σ (E ) = (Se ) n σ (E n ) = (Se n ) n.

3 98 C. Tong et al. / Discrete Applied Matheatics 7 (009) Fig... σ (C C n ) for n {{ } { } { } {6 6} {7 7}}. Case.. n od. σ (V) = (Sv ) n 6 σ (E ) = (Se ) n 6 σ (E n ) = (Se n ) n 6 Case.. n od. σ (V) = (Sv ) n σ (E ) = (Se ) n σ (E n ) = (Se n ) n Case.. n od. σ (V) = (Sv ) n σ (E ) = (Se ) n σ (E n ) = (Se n ) n β β β β β β β β β β β β β β β β β. β β β β β β. β β β β β β β β β.

4 C. Tong et al. / Discrete Applied Matheatics 7 (009) Case.. n od. σ (V) = (Sv ) n σ (E ) = (Se ) n σ (E n ) = (Se n ) n Case. od. β β β β β β β β β β β β. Sv = β 6 β 6 β 6 β 6 β 6 Se = β 6 β 6 β 6 β 6 β 6 Se n = β 6 β 6 β 6 β 6 β 6. Case.. n od (see Fig..()). σ (V) = (Sv ) n 6 6 β σ (E ) = (Se ) n 6 σ (E n ) = (Se n ) n 6 Case.. n od. σ (V) = (Sv ) n 7 β 6 β 6 β 6 β 6 β 6 6 β 6 β 6 β σ (E ) = (Se ) n 7 σ (E n ) = (Se n ) n 7 Case.. n od. Case... n =. β 6 β 6 β 6 β 6 β 6 β 6 β 6 β 6 β 6 β 6. β 6 β 6 β 6 β 6 β 6 β 6 6 β 6 β β 6 β 6 β 6 β 6 β 6 β 6 β 6 β 6 β 6 β 6 β 6 β 6. σ (V) = β 6 β 6 β 6 σ (E ) = () 6 () 6 () 6 σ (E n ) = () 6 () 6 () 6. Case... n 8. σ (V) = (Sv ) n 8 6 β β 6 β 6 β 6 β 6 β 6 β 6 β 6 σ (E ) = (Se ) n 8 6 β β 6 β 6 β 6 β 6 β 6 β 6 β 6 σ (E n ) = (Se n ) n 8 6 β β 6 β 6 β 6 β 6 β 6 β β 6. Case.. n od. σ (V) = (Sv ) n 6 β σ (E ) = (Se ) n σ (E n ) = (Se n ) n β 6 β 6 β 6 β 6 β β 6 β 6 β 6 β 6 β 6 β 6.

5 600 C. Tong et al. / Discrete Applied Matheatics 7 (009) Case. od. Sv = β β 7 β 7 β 7 β Se = β 7 β 7 β β 7 β 7 Se n = β 7 β 7 β 7 β 7 β. Case.. n od (see Fig..()). σ (V) = (Sv ) n 7 σ (E ) = (Se ) n 7 σ (E n ) = (Se n ) n 7 Case.. n od. Case... n =. σ (V) = β β β 7 7 β 7 β β 7 β 7 β 7 β β 7 β β 7 β 7 β β 7 β β 7 β 7 β 7 β β 7 σ (E ) = β 7 β 7 β 7 σ (E n ) = β Case... n 8. β 7 β 7. σ (V) = (Sv ) n 8 β β 7 β 7 β 7 β 7 β 7 β 7 β σ (E ) = (Se ) n 8 7 β β 7 σ (E n ) = (Se n ) n 8 Case.. n od. Case... n =. 7 β β 7 σ (V) = β 7 β β 7 β 7 β 7 β 7 β β 7 β 7 β 7 β 7 β β 7 β. β 7 β σ (E ) = β 7 β 7 β 7 β 7 σ (E n ) = β Case... n 9. β 7 σ (V) = (Sv ) n β 7 β 7. 7 β σ (E ) = (Se ) n σ (E n ) = (Se n ) n Case. od. β β β 7 7 β β β β 7 β β 7 β β Sv = β β β β β Se = β β β β β Se n = β β β β β.. β 7 β 7 β.

6 C. Tong et al. / Discrete Applied Matheatics 7 (009) Case.. n od (see Fig..()). σ (V) = (Sv ) n β σ (E ) = (Se ) n σ (E n ) = (Se n ) n Case.. n od. σ (V) = (Sv ) n β β β β β σ (E ) = (Se ) n σ (E n ) = (Se n ) n β β β β. β β β β β β β β β β β. Case. od and n od (see Fig..()). Sv = β β β β β Se = β β β β β Se n = β β β β β. σ (V) = (Sv ) n β σ (E ) = (Se ) n σ (E n ) = (Se n ) n β β β β β β β β β β β. Clearly σ is a -total coloring of C C n. Now we take Case for exaple to verify that σ is an equitable total -coloring of C C n. Let T(C C n ) = (T T T T T ) where T i denotes the i-th color class of σ. Since T = T = T = (n+)/ and T = T = (n )/ we have T i T j ( i < j ) for Case. Siilarly we have T i T j ( i < j ) for all other Cases. Hence σ is an equitable total -coloring of C C n. Acknowledgeents We are very grateful to the referees for their careful reading with corrections and useful coents. References [] J.L. Baril H. Kheddouci O. Togni Adjacent vertex distinguishing edge-colorings of eshes Australas. J. Cobin. (006) [] M. Behzad Graphs and their chroatic nubers Ph.D. Thesis Michigan State University 96. [] C.N. Capos C.P. de Mello A result on the total colouring of powers of cycles Discrete Appl. Math. (007) [] K. Edwards M. Horňák M. Woźniak On the neighbour-distinguishing index of a graph Graphs Cobin. () (006) 0. [] Hung-lin Fu Soe results on equalized total coloring Congr. Nuer. 0 (99) 9. [6] A.J.W. Hilton Ji-ping Liu Cheng Zhao The total chroatic nubers of joins of sparse graphs Australas. J. Cobin. 8 (00) 9 0. [7] A. Kenitz M. Marangio Total colorings of Cartesian products of graphs Congr. Nuer. 6 (00) [8] A.V. Kostochka The total coloring of a ultigraph with axial degree Discrete Math. 7 (977) 6 6. [9] Guang-rong Li Li-in Zhang Total chroatic nuber of one kind of join graphs Discrete Math. 06 (006) [0] C.J.H. McDiarid A. Sánchez-Arroyo Deterining the total colouring nuber is NP-hard Discrete Math. (99) 6. [] M.A. Seoud Total chroatic nubers Appl. Math. Lett. (6) (99) 7 9. [] M.A. Seoud A.E.I. Abd el Maqsoud R.J. Wilson J. Willias Total colourings of Cartesian products Int. J. Math. Educ. Sci.Technol. 8 (997) [] A. Sánchez-Arroyo Total colouring regular bipartite graphs is NP-hard Discrete Math. 78 (989) 9. [] V.G. Vizing Soe unsolved probles in graph theory Uspehi Mat. Nauk. (968) 7. [] Shu-dong Wang Shan-chen Pang The deterination of the total chroatic nuber of series-parallel graphs with (G) Graphs Cobin. (00) 0. [6] Wei-fan Wang Equitable total coloring of graphs with axiu degree Graphs Cobin. 8 (00) [7] Wei-fan Wang The total chroatic nuber of planar graphs with axiu degree ten J. Graph theory. (007) 9 0. [8] Yi-xian Yang hua-ping Liu Fangc-hun Yang Zhong-fu Zhang Total chroatic nuber of Cartesian products Math. Appl. () (999) 08. [9] Zhong-fu Zhang On the equitable total colorings of soe join graphs J. Infor. Coput Sci. () (00) [0] Zhong-fu Zhang Xiang-en Chen Jing-wen Li Bian Liang Peng-xiang Qiu A note on the relation of adjacent strong edge coloring and total coloring of graphs 006 (subitted for publication). [] B. Zazek J. Zerovnik Behzad-Vizing conjecture and Cartesian product graphs Electron. Notes Discrete Math. 7 (00) [] M. Zwonek A note on total colourings of digraphs Discrete Math. 06 (006) 8 9.