On the Total Neighbor Sum Distinguishing Index of IC-Planar Graphs

Donghan Zhang,Fugang Chao,Chao Li

doi:10.3390/sym13101787

Abstract

A proper total k-coloring ϕ of G with ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each uv∈E(G) is called a total neighbor sum distinguishing k-coloring, where EG(u)={uv|uv∈E(G)}. Pilśniak and Woźniak conjectured that every graph with maximum degree Δ exists a total neighbor sum distinguishing (Δ+3)-coloring. In this paper, we proved that any IC-planar graph with Δ≥12 satisfies this conjecture, which improves the result of Song and Xu.

Highlights

We proved that any IC-planar graph with ∆ ≥ 12 satisfies this conjecture, which improves the result of Song and Xu
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We proved that any IC-planar graph with ∆ ≥ 12 satisfies the tnsd index conjecture by the discharging method and Combinatorial Nullstellensatz

Summary

Introduction

A total neighbor sum distinguishing k-coloring (for short, tnsd) is a mapping φ : T ( G ) → {1, 2, · · · , k} satisfying the following conditions:. In 2015, Pilśniak and Woźniak [1] first studied the coloring and gave the conjecture about the tnsd index as follows: Conjecture 1 ([1]). The tnsd index on IC-planar graphs has been extensively studied. (2) IC-planar graph with ∆ ≥ 14 but without 2-vertex incident with crossed edge ([8]). There is a result of list version about the tnsd index of IC-planar graphs, see [10]. We discuss any IC-planar graph with maximum degree ∆ ≥ 12 and obtain the following result, which extends the result of (3) in Theorem 2.

Preliminaries

Proof of Theorem 3

Conclusions