Total Coloring of the Generalized Lexicographic Product of Some Graphs
Sun Xiang-tao · Journal of Shanxi University · 2013
Suppose that G is a wheel,or fan,or star with vertex set {t0,t1,…,tn-1},where t0 is the vertex with maximum degree and n≥5.Let G be the generalized lexicographic product of graph G and a sequence of vertex disjoint graphs hn=(Hi)i∈{0,1,…,n-1},where each Hi is a simple graphs with m vertices.The following results are obtained:(1)If H0 is the complement of a complete graph,then the total chromatic number of graph G is(n-1)m+1;(2)If H0 is a complete graph,then the total chromatic number of graph G is mn;(3)If H0 is a bipartite graph,then the total chromatic number of graph G is Δ(H0)+(n-1)m+1,where Δ(H0) denotes the maximum degree of H0;(4)If H0 is a cycle,then the total chromatic number of graph G is(n-1)m+3.