Neighbor sum distinguishing total chromatic number of K4-minor free graph

摘要

A k-total coloring of a graph G is a mapping φ:V(G) ∪ E(G) →{1,2,...,k} such that no two adjacent or incident elements in V(G) ∪ 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 φ is a k-neighbor sum distinguishing total coloring of G if f(u) ≠ f(v) for each edge uv ∈ E(G).Denoteχ"Σ(G) the smallest value k in such a coloring of G.Pil(s)niak and Wo(z)niak conjectured that for any simple graph with maximum degree △(G),χ"Σ(G) ≤△(G) + 3.In this paper,by using the famous Combinatorial Nullstellensatz,we prove that for K4-minor free graph G with △(G) ≥ 5,χ"Σ(G)=△(G)+1 if G contains no two adjacent △-vertices,otherwise,χ"Σ(G)=△(G) + 2.