Neighbor sum distinguishing total chromatic number of K (4)-minor free graph

摘要

A k-total coloring of a graph G is a mapping I center dot: V (G) ai integral E(G) -> {1; 2,..., k} such that no two adjacent or incident elements in V (G) ai integral E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v: We say that I center dot is a k-neighbor sum distinguishing total coloring of G if f(u) 6 not equal f(v) for each edge uv a E(G): Denote chi (I ) pound (aEuro3) (G) the smallest value k in such a coloring of G: PilA > niak and WoAniak conjectured that for any simple graph with maximum degree Delta(G), chi (I ) pound (aEuro3) ae Delta(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K (4)-minor free graph G with Delta(G) > 5; chi (I ) pound (aEuro3) = Delta(G) + 1 if G contains no two adjacent Delta-vertices, otherwise, chi (I ) pound (aEuro3) (G) = Delta(G) + 2.