The Equitable Total Chromatic Number of Some Join graphs

Gang MA, Ming MA

College of Mathematics and Computer Science Northwest University for Nationalities Lanzhou, Gansu, China 730030.

DOI: 10.4236/ojapps.2012.24B023                

Abstract

A proper total-coloring of graph G is said to be equitable if the number of elements (vertices and edges) in any two color classes differ by at most one, which the required minimum number of colors is called the equitable total chromatic number. In this paper, we prove some theorems on equitable total coloring and derive the equitable total chromatic numbers of Pm V Sn, Pm V Fn and Pm V Wn.

Keywords

join graph; equitable total coloring; equitable total chromatic numbers

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	H. P. Yap, Total Colorings of Graphs, Berlin: Lecture Notes in Mathematics, 1623, Springer, 1996.
[2]	J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, New York: The Macmillan Press Ltd, 1976.
[3]	Tian Feng, MA Zhong-fan, Graph Theory and Network Flow Theory, Beijin: Science Press, 1987.
[4]	Zhang Zhong-fu, Wang Jian-fang, The Progress of Total-Colouring of Graphs, Advances in Mathematics, 1992,21(4):390-397.
[5]	Zhang Zhong-fu, Zhang Jian-xun, On Some Sufficient Conditions of First Kind Graph, Journal of Mathematics, 1985,5(2):161-165.
[6]	Ma Gang, Zhang Zhong-fu, On the Equitable Total Coloring of Multiple Join-graph, Journal of Mathematical Research and Exposition, 2007,27(2):351-354.
[7]	Wang Wei-fan, Equitable Total Coloring of Graphs with Maximum Degree 3, Graphs and Combin, 2002,18:677-685.
[8]	Ma Gang, Zhang Zhong-fu, Qiang Hui-ying, On equitable total chromatic number of Cm V Fn, Journal of Lanzhou Jiaotong University, 2005,24(4):147-149.
[9]	Ma Gang, Zhang Zhong-fu, on Adjacent Vertex-distinguishing-equitable Total Coloring of Double Graphs, J. of Jilin University (Science Edition), 2009, 47(6): 1160-1164.
[10]	Zhang Zhong-fu, Li Mu-chun, Yao Bin, et al, On the Vertex Distinguishing Equitable Edge-coloring of Graphs, ARS Combinatoria, 2008, 86: 193-200.
[11]	Zhang Zhong-fu, Li Jing-wen, Zhao Chuan-cheng, et al, On the Vertexdistinguishing-equitable Edge Chromatic of some Join-graphs, Acta Mathematica Sinica, Chinese Series, 2007, 50(1): 197-204.