A note on the neighbor sum distinguishing total coloring of planar graphs

Abstract

Listen

Translate article

Let G=(V(G),E(G)) be a graph and ϕ be a proper total k-coloring of G. Let f(v) denote the sum of the color on a vertex v and colors on all the edges incident with v. ϕ is neighbor sum distinguishing if f(u)≠f(v) for each edge uv∈E(G). The smallest integer k for which such a coloring of G exists is the neighbor sum distinguishing total chromatic number and denoted by χΣ″(G). Pilśniak and Woźniak conjectured that for any simple graph with maximum degree Δ(G), χΣ″(G)≤Δ(G)+3. It is known that for any simple planar graph, χΣ″(G)≤max⁡{Δ(G)+3,14} and χΣ″(G)≤max⁡{Δ(G)+2,16}. In this paper, by using the famous Combinatorial Nullstellensatz, we show that for any simple planar graph, χΣ″(G)≤max⁡{Δ(G)+2,14}. The bound Δ(G)+2 is sharp.