Some Results Of The Edge-cocoloring Of Graphs

In 1980,Lowell Beineke and Richard Ringeisen[1] introduced the concept of the cochro-matic index of a graph.The cochromatic index of G,denoted by z' (G) , is the minimumnumber of subsets into which the edge set of G can be partitioned so that each subsetis an independent set of edges,a K3,or a star in G.In 2006,LIU Xin-sheng[7] introducedthe concept of the edge-cochromatic number of G,denoted by z' (G),is the minimum num-ber of subsets into which the edge set of G can be partitioned so that each subset is anindependent set of edges or a star in G.In this paper,we mainly study the triangle-free cochromatic index of G based on theboundΦ(G)≤z' (G)≤Φ(G) + 1.Here the triangle-free cochromatic index namely is theedge-cochromatic number.Our paper have four sections:1.We will introduce the concepts and the lemmas associating with the edge-cocoloringof graphs.2.We will discuss the classification of graphs.Based on the edge-cochromatic number,we will divide graphs into two kinds .Some simple determination criterion and examplesgraphs of each class will be presented.3.We will study the edge-cocoloring of several kinds join-graphs,and give the edge-cochromatic number of S_m∨S_n,S_m∨W_n,S_m∨P_n and S_m∨C_n.4.We will introduce the concept of the uniquely edge-cocolourable graphs and give itssimple properties.At the same time,we will construct three di?erent types of the uniquelyedge-cocolourable graph.