Neighbor sum distinguishing total choosability of planar graphs

Abstract

A total-k-coloring of a graph G is a mapping $$c: V(G)\cup E(G)\rightarrow \{1, 2,\dots , k\}$$c:V(G)źE(G)ź{1,2,ź,k} such that any two adjacent or incident elements in $$V(G)\cup E(G)$$V(G)źE(G) receive different colors. For a total-k-coloring of G, let $$\sum _c(v)$$źc(v) denote the total sum of colors of the edges incident with v and the color of v. If for each edge $$uv\in E(G)$$uvźE(G), $$\sum _c(u)\ne \sum _c(v)$$źc(u)źźc(v), then we call such a total-k-coloring neighbor sum distinguishing. The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by $$\chi _{\Sigma }^{''}(G)$$źΣźź(G). Pilśniak and Woźniak conjectured $$\chi _{\Sigma }^{''}(G)\le \Delta (G)+3$$źΣźź(G)≤Δ(G)+3 for any simple graph with maximum degree $$\Delta (G)$$Δ(G). In this paper, we prove that for any planar graph G with maximum degree $$\Delta (G)$$Δ(G), $$ch^{''}_{\Sigma }(G)\le \max \{\Delta (G)+3,16\}$$chΣźź(G)≤max{Δ(G)+3,16}, where $$ch^{''}_{\Sigma }(G)$$chΣźź(G) is the neighbor sum distinguishing total choosability of G.