| Abstract: | With the definition of proper total coloring of a graph, an Adjacent Vertex-Distinguishing Total Coloring (AVDTC) means that none of the two adjacent vertices are incident with the same set of colors. The concept of the AVDTC is proposed by Zhongfu Zhang (2004), and the AVDTC of graphs such as path, cycle, complete graph, complete bipartite graph, star and tree are discussed in Zhang's paper. The AVDTC of Pm×Pn , Pm×Cn , Cn×Cn are also given where Pm , Cn are a denoted path with order m and a circle with order n,respectively; the AVDTC of Mycielski graph of some graphs such as path, circle and so on are given in another Zhang's paper (2000). For the adjacent vertex-distinguishing total chromatic number, a conjecture is given in Zhang's paper (2004). Let G（V, E） be a simple connected graph of order n（n≥2）, k be a natural number and f be a k-adjacent vertex-distinguishing total coloring of graph G. If f satisfies the condition ||Vi∪Ei|－|Vj∪Ej||≤1 （i≠j）, where Vi∪Ei={v|f（‘v）=i}∪{e|f（e）=i}, C（i）=Vi∪Ei, then f is called an equitableadjacent vertex-distinguishing total coloring of graph G（k-EAVDTC） and χeat（G）=min{k∣G has k-EAVDTC} is called the chromaticnumber of the equitable adjacent-distinguishing total coloring of graph G. This paper gives the equitable adjacent vertex distinguishing total coloring chromatic number of path Pn and cycle Cn and graph Kt3 andgraph Dm,4 and gear wheel ■n. |
