某些中间图的邻点可区别E-全色数

设G（V,E）是简单连通图,T（G）为图G的所有顶点和边构成的集合,并设C是k-色集（k是正整数）,若T（G）到C的映射f满足：对任意uv∈E（G）,有f（u）≠f（v）,f（u）≠f（uv）,f（v）≠f（uv）,并且C（u）≠C（v）,其中C（u）={f（u）}∪{f（uv）｜uv∈E（G）}.那么称f为图G的邻点可区别E-全染色（简记为k-AVDETC）,并称χ_（at）~e（G）=min{k｜图G有k-邻点可区别E-全染色}为G的邻点可区别E-全色数.图G的中间图M（G）就是在G的每一个边上插入一个新的顶点,再把G上相邻边上的新的顶点相联得到的.探讨了路、圈、扇、星及轮的中间图的邻点可区别E-全染色,并给出了这些中间图的邻点可区别E-全色数.

Adjacent Vertex-distinguishing E-Total Chromatic Numbers of Some Middle Graphs

Let G（V, E） be a simple graph and T（G） be the set of vertices and edges of G. Let C be a k-color set（k is a positive integer）. An E-total k-coloring f of G is a function f： T（G） →4 C such that neither adjacent vertices of V（G） nor incident elements between vertices and edges of T（G） receive the same color. For any u e V（G）, denote C（u） → {f（u）} U{f（uv）uv ∈ E（G）}. The E-total k-coloring f of G is called the adjacent vertex-distinguishing if C（u） → C（v） for any edge uv ∈ E（G）. And the smallest number of the colors required in an adjacent vertex distinguishing E-total coloring is called the adjacent vertex-distinguishing E-total chromatic number Xat （G） of G. Based on a simple graph G, the middle graph M（G） of a graph G is the graph obtained from G by inserting a new vertex into every edge of G and by joining by edges those pairs of these new vertices which lie on adjacent edges of G. In this paper, we investigate the adjacent vertex distinguishing E-total coloring of middle graphs of some simple graphs such as path, cycle, fan, star, wheel, and we find that the exact value of the adjacent vertex distinguishing E-total chromatic numbers of M（Pn） and M（Cn） axe both 4, and that of M（Sn）, M（Fn） and M（Wn） are all n ＋ 1.