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

Abstract

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). Denote {{\displaystyle X}}_{\mathrm{\Sigma }}^{\text{'}\text{'}}(G) the smallest value k in such a coloring of G. Pilśniak andWoźniak conjectured that for any simple graph with maximum degree Δ(G), {{\displaystyle X}}_{\Sigma }^{\text{'}\text{'}}(G)\le \mathrm{\Delta }(G)+\mathrm{3}. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K₄-minor free graph G with Δ(G)≥5, {{\displaystyle X}}_{\mathrm{\Sigma }}^{\text{'}\text{'}}(G)=\mathrm{\Delta }(G)+\mathrm{1} if G contains no two adjacent Δ-vertices, otherwise, {{\displaystyle X}}_{\mathrm{\Sigma }}^{\text{'}\text{'}}(G)=\mathrm{\Delta }(G)+\mathrm{2}.

Keywords

Neighbor sum distinguishing total coloring / Combinatorial Nullstellensatz / K₄-minor free graph

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