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

Donghan Zhang
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Donghan Zhang: School of Mathematics and Statistics and Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an 710129, China

Mathematics, 2021, vol. 9, issue 7, 1-11

Abstract: A theta graph ? 2 , 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 ? of G is a proper total coloring of G such that ? z ? E G ( u ) ? { u } ? ( z ) ? ? z ? E G ( v ) ? { v } ? ( z ) for each edge u v ? E ( G ) , where E G ( u ) denotes the set of edges incident with a vertex u . In 2015, Pil?niak and Wo?niak introduced this coloring and conjectured that every graph with maximum degree ? admits an NSD total ( ? + 3 ) -coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree ? ? 9 but without theta graphs ? 2 , 1 , 2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

Keywords: IC-planar graphs; neighbor sum distinguishing total choosibility; Combinatorial Nullstellensatz (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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