Adjacent Vertex-distinguishing E-total Coloring on Some Join Graphs Cm∨Gn

摘要

Let G(V,E) be a simple connected graph and k be positive integers.A mapping f from V ∪ E to {1,2,...,k} is called an udjacent vertex-distinguishing E-total coloring of G(abbreviated to k-AVDETC),if for (V)uv ∈ E(G),wehave f(u) ≠ f(v),f(u) ≠f(uv),f(v) ≠ f(uv),C(u) ≠ C(v),where C(u) ={f(u)}∪{f(uv)|uv ∈ E(G)}.The least number of k colors required for which G admits a k-coloring is called the adjacent vertex-distinguishing E-total chromatic number of G is denoted by xeat(G).In this paper,the adjacent vertex- distinguishing E-total colorings of some join graphs Cm ∨ Gn are obtained,where Gn is one of a star Sn,a fan Fn,a wheel Wn and a complete graph Kn.As a consequence,the adjacent vertex-distinguishing E-total chromatic numbers of Cm ∨ Gn are confirmed.