Some Equal Degree Graph Edge Chromatic Number

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.