The Sigma Edge Chromatic Number Of A Graph

There have been numerous studies using a variety of methods for the purpose of uniquely identifying (or distinguishing) the vertices of a graph or the edges of a graph. Many of these methods have involved graph colorings such as edge-distinguishing vertex coloring, vertex-distinguishing edge coloring, vertex coloring edge weights. More generally, All the colorings which have been mentioned above is called neighbor-distinguishing coloring, a vertex coloring or edge coloring c of a graph G is neighbor-distinguishing if c gives rise to a edge labeling or vertex labeling in which every pair of adjacent edges or vertices in G are assigned distinct labels. Prof. Chartrand introduced a new neighbor-distinguishing vertex coloring of a graph called sigma coloring, and several good results have been presented.This paper focuses on the sigma edge coloring, and the main works as follows:(1) The concept of the sigma edge coloring is proposed, and the relationship between sigma edge coloring and edge coloring, edge sizes is studied.①Relationships between the sigma edge chromatic number and the edge chromatic number of a graph:First, it is turned out to be that the sigma edge chromatic number of a graph never exceeds its edge chromatic number.Then, it is proved that for every pair a and b of positive integers with a≤b, there is a connected graph G withσ'(G)=a andχ'(G)=b.②Relationships between the sigma edge chromatic number and the size of a graph:First, it is turned out to be that there is no nontrivial connected graph G of sizeεwithσ(G)=ε-1.Then, it is proved that k andεare positive integers with k≤ε≤2k, there exists a connected graph G of sizeεwithε(G)=kif and only if k≠n-1.(2) The concept of sigma edge continuous is proposed. The sigma edge chromatic number of path, cycle and star are given by the relationship between sigma edge coloring and edge coloring. Then the sigma edge continuous of road, cycle and star are proved by constructing coloring schemes.(3) According to the structure characteristics of wheel graph, windmill graph, fully n-ary tree graphs, comb and caterpillar tree and by constructing coloring schemes, the upper bounds sigma chromatic number of wheel graph is studied, Then the sigma chromatic number of windmill graph is given and it is turned out to be that windmill graph is not sigma edge continuous. Finally, the sigma chromatic number of fully n-ary tree graphs, comb and caterpillar tree is given and it is proved that these graph are sigma edge continuous.