Neighbor sum distinguishing total choosability of planar graphs without 4-cycles adjacent to 3-cycles

Abstract

Let φ be a proper total coloring of a graph G with integers as colors. For a vertex v, let w(v) denote the sum of colors assigned to edges incident to v and the color assigned to v. If w(u)≠w(v) whenever uv∈E(G), then φ is called a neighbor sum distinguishing total coloring. A k-assignment L of G is a list assignment L of integers to vertices and edges with |L(z)|=k for each z∈V(G)∪E(G). A total-L-coloring is a total coloring φ of G such that φ(v)∈L(v) whenever v∈V(G) and φ(e)∈L(e) whenever e∈E(G). The smallest integer k such that G has a neighbor sum distinguishing total-L-coloring for every k-assignment L is called the neighbor sum distinguishing total choosability of G and is denoted by Ch ∑ ’’(G). Wang, Cai, and Ma [15] proved that every planar graph G without 4-cycles with ∆(G)≥7 has Ch ∑ ’’(G)≤∆(G)+3. In this work, we strengthen the result of Wang et al by proving that Ch ∑ ’’(G)≤∆(G)+3 for every planar graph G without 4-cycles adjacent to 3-cycles with ∆(G)≥7.