Neighbor sum distinguishing total colorings of planar graphs with girth at least 5

Abstract

Let G = (V, E) be a graph and φ be a proper k-total coloring of G. For a vertex v of G, let f(v) = Σ_uvϵE(G) φ(uv) + φ(v)- The coloring φ is neighbor sum distinguishing if f(u) ≠ f(v) for each edge uv ϵ E(G). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by x"_Σ(G)- By using the famous Combinatorial Nullstellensatz, we determine X"_σ(G) for any planar graph G with girth at least 5 and Δ(G) ≥ 7.