Some Equal Degree Graph Edge Chromatic Number

Jun Li ,Cai Hui Lan,Qiu Ju Yue,Zhong Yi Feng,Zhi Guo Ren

doi:10.1051/matecconf/20164401044

Abstract

Let G(V, E) be a simple graph and k is a positive integer, if it exists a mapping of f, and satisfied with f(e1)≠6 = f(e2) for two incident edges e1,e2∉E(G), f(e1)≠6=f(e2), then f is called the k-proper-edge coloring of G(k-PEC for short). The minimal number of colors required for a proper edge coloring of G is denoted by X′ (G) and is called the proper edge chromatic number. It exists a k-proper edge coloring of simple graph of G, for any two adjacent vertices u and v in G, the set of colors assigned to the edges incident to u differs from the set of colors in cident to v, then f is called k-adjacent-vertex-distinguishing proper edge coloring, is avvreviated k-AVDEC, also called a adjacent strong edge coloring. The minimal number of colors required for a adjacent-vertex-distinguishing edge coloring of G is denoted by X′ad (G) and is called adjacent-vertex-distinguishing edge chromatic number. The new class graphs of equal degree graph are be introduced, and this class graphs adjacent-vertex-distinguishing edge chromatic numbers of path, cycle, fan, complete graph, wheel, star are presented in this paper.

Highlights

The coloring problem of graphs is a extremely difficult problem, widely applied in practice
Some network problems can be converted to the edge coloring [2 9] and adjacent strong edge coloring
V (Fn ) {u0,u1,u2,un}, V c {v0, v1, v2,un}, there are three cases to be discussed as follow: Case1 When n=2,M(F2) =M(C3), so we can obtain X c(M (F3)) X c(M (C3)) 6, the conclusion is true

Summary

Introduction

The coloring problem of graphs is a extremely difficult problem, widely applied in practice. Some network problems can be converted to the edge coloring [2 9] and adjacent strong edge coloring. Definition 1 [10] For a graphG(V, E), if a proper coloring f is satisfied with C(u) z C(v) for u,v V (G)(u z v) and uv E(G) f is called k-adjacent-vertex- distinguishing edge coloring of G, is abbreviated k-AVDEC, and. Is called the adjacent-vertex-distinguishing edge chromatic number of G [6] ˈwhere C(u) { f (uv) uv E(G). Aauthore-mail: 527876625@.qq.com This studywas supported by Lanzhou City University Ph.D. Research Fund (21265099,41361013,GS[2013]GHB1084, LZCU-BS2013-09 and LZCU-BS2013-12).

Proof Let

The we have