Neighbor sum distinguishing total colorings of K₄-minor free graphs

Abstract

A total [k]-coloring of a graph G is a mapping ϕ: V (G) ∪E(G) → {1,2, ..., k} such that any two adjacent elements in V (G)∪E(G) receive different colors. Let f(v) denote the sum of the colors of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uv ∈E(G),f(u)\ne f(v). By {\chi }_{nsd}^{″}(G), we denote the smallest value k in such a coloring of G. Pilśniak and Woźniak conjectured {\chi }_{nsd}^{″}(G)\le ▵(G)+\mathrm{3} for any simple graph with maximum degree Δ(G). This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for K4-minor free graphs. Furthermore, we show that if G is a K4-minor free graph with ▵(G)\ge \mathrm{4}, then {\chi }_{nsd}^{″}(G)\le ▵(G)+\mathrm{2} The bound Δ(G) + 2 is sharp.

Key words： K₄-minor free graph; neighbor sum distinguishing (nsd)

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