Neighbor product distinguishing total colorings of planar graphs with maximum degree at least ten

Abstract

A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2,. .. , k}. Let p(u) denote the product of the color on a vertex u and colors on all the edges incident with u. For each edge uv ∈ E(G), if p(u) = p(v), then we say the coloring c distinguishes adjacent vertices by product and call it a neighbor product distinguishing k-total coloring of G. By χ (G), we denote the smallest value of k in such a coloring of G. It has been conjectured by Li et al. that ∆(G) + 3 colors enable the existence of a neighbor product distinguishing total coloring. In this paper, by applying the Combinatorial Nullstellensatz, we obtain that the conjecture holds for planar graph with ∆(G) ≥ 10. Moreover, for planar graph G with ∆(G) ≥ 11, it is neighbor product distinguishing (∆(G) + 2)total colorable, and the upper bound ∆(G) + 2 is tight.