Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

Donghan Zhang

doi:10.3390/math9070708

Abstract

A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total (Δ+3)-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ≥9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

Highlights

The graphs mentioned in this paper are finite, undirected, and simple
Qu et al [10] proved that this conjecture holds for any planar graph G with maximum degree ∆(G) ≥ 13
We reduce the condition ∆ ≥ 10 of (3) in Theorem 1 to ∆ ≥ 9 and obtain the list version result as follows

Summary

Introduction

The graphs mentioned in this paper are finite, undirected, and simple. For undefined terminology and notations, here we follow [1]. The smallest integer k such that G has an NSD total L-coloring for any k-list total assignment L is called the NSD total choice number of G, denoted by chtΣ(G). Qu et al [10] proved that this conjecture holds for any planar graph G with maximum degree ∆(G) ≥ 13. Wang et al [11] confirmed Conjecture 2 for every planar graph G with maximum degree ∆(G) ≥ 8 but without theta graphs Θ2,1,2. Song et al [12] discussed the NSD total L-coloring of any IC-planar graph and obtained the following theorem. In 1999, Alon developed a general algebraic technique that is called Combinatorial Nullstellensatz It has numerous applications in additive number theory, combinatorics, and graph coloring problems.

Proof of Theorem 3

Suppose that the configuration of