Equitable total coloring of CmCn

The equitable total chromatic number of a graph G is the smallest integer k for which G has a k-total coloring such that the number of vertices and edges colored with each color differs by at most one. In this paper, we show that the Cartesian product graphs of C"m and C"n have equitable total 5-coloring for all m>=3 and n>=3.

[1]  Jingwen Li,et al.  A Note on Adjacent Strong Edge Coloring of K(n,m) , 2006 .

[2]  Alexandr V. Kostochka,et al.  The total coloring of a multigraph with maximal degree 4 , 1977, Discret. Math..

[3]  Anthony J. W. Hilton,et al.  The total chromatic numbers of joins of sparse graphs , 2003, Australas. J Comb..

[4]  M. Seoud,et al.  Total chromatic numbers , 1992 .

[5]  J Hang,et al.  THE TOTAL CHROMATIC NUMBER OF SOME GRAPHS , 1988 .

[6]  Frank Harary,et al.  Graph Theory , 2016 .

[7]  Xiangen Chen,et al.  AVDTC numbers of generalized Halin graphs with maximum degree at least 6 , 2008 .

[8]  Abddn SANCHEZ-ARROYO,et al.  Determining the total colouring number is np-hard , 1989, Discret. Math..

[9]  C. N. Campos,et al.  A result on the total colouring of powers of cycles , 2004, Discret. Appl. Math..

[10]  Olivier Togni,et al.  Adjacent vertex distinguishing edge-colorings of meshes , 2006, Australas. J Comb..

[11]  Weifan Wang.  Total chromatic number of planar graphs with maximum degree ten , 2007 .

[12]  A. E. I. Abd el Maqsoud,et al.  Total colourings of Cartesian products , 1997 .

[13]  V. G. Vizing.  SOME UNSOLVED PROBLEMS IN GRAPH THEORY , 1968 .

[14]  Wei-Fan Wang.  Equitable Total Coloring of Graphs with Maximum Degree 3 , 2002, Graphs Comb..

[15]  Keith Edwards,et al.  On the Neighbour-Distinguishing Index of a Graph , 2006, Graphs Comb..

[16]  Janez Zerovnik,et al.  Behzad-Vizing conjecture and Cartesian product graphs , 2002, Electron. Notes Discret. Math..

[17]  Shu-Dong Wang,et al.  The Determination of the Total Chromatic Number of Series-Parallel Graphs with (G) ≥ 4 , 2005, Graphs Comb..

[18]  Malgorzata Zwonek.  A note on total colourings of digraphs , 2006, Discret. Math..

[19]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[20]  Colin McDiarmid,et al.  Total colouring regular bipartite graphs is NP-hard , 1994, Discret. Math..

[21]  Limin Zhang,et al.  Total chromatic number of one kind of join graphs , 2006, Discret. Math..