Neighbor sum distinguishing total chromatic number of K ₄-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) 6 ≠ f(v) for each edge uv ∈ E(G): Denote χ ^″_Σ (G) the smallest value k in such a coloring of G: Pilśniak and Woźniak conjectured that for any simple graph with maximum degree Δ(G), χ ^″_Σ ≤ Δ(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K ₄-minor free graph G with Δ(G) > 5; χ ^″_Σ = Δ(G) + 1 if G contains no two adjacent Δ-vertices, otherwise, χ ^″_Σ (G) = Δ(G) + 2.