Neighbor Full Sum Distinguishing Total Coloring of Planar Graphs with Girth at Least 5 - Zhongzheng Yue; Fei Wen (Axioms)
A neighbor full sum distinguishing total coloring of a graph G is a proper k-total coloring <semantics>ϕ</semantics> such that no two adjacent vertices u and v satisfy <semantics>ω(u)≠ω(v)</semantics>, where <semantics>ω(v)=ϕ(v)+∑e∋vϕ(e)+∑u∈N(v)ϕ(u)</semantics> and <semantics>N(v)={u|uv∈E(G)}</semantics>. The minimum positive integer k is called the neighbor full sum distinguishing total coloring of G, or <semantics>ftndiΣ(G)</semantics> for short. In this article, we verify that for a normal planar graph G, <semantics>ftndiΣ(G)≤Δ(G)+3</semantics> if <semantics>g(G)≥5</semantics> and <semantics>Δ(G)≥22</semantics>, which extends the result of Yue, et al. by reducing the girth condition from 6 to 5.