Let f:E(G)→ ｛ 1,2,…,k} be a non-proper k-edge coloring of G,and 1≤k≤Δ.If for any two adjacent vertices u,v ∈V(G)with d(u)=d(v)satisfy C(u)=C(v),f is called a k-vertex-reducible edge coloring,where C(u)denotes the set of colors of edges incident with u.The maximum positive integer k is called vertex-reducible edge chromatic number of G.According to the characters of the lexicographic product graphs,we apply combinatorial analysis to give a lower bound of the vertex reducible edge chromatic number of the lexicographic product G[H]for simple graphs G and H.As applications,the vertex-reducible edge chro-matic numbers of Kn[K2m],Kn[H]and Pn[H]are obtained.
lexicographic productvertex reducible edge coloringvertex reducible edge chromatic number
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