Neighbor Sum Distinguishing Total Chromatic Number of Graphs with Lower Average Degree
Year: 2023
Author: Danjun Huang, Dan Bao
Journal of Mathematical Study, Vol. 56 (2023), Iss. 2 : pp. 206–218
Abstract
For a given simple graph G=(V(G),E(G)), a proper total-k-coloring c:V(G)∪E(G)→{1,2,...,k} is neighbor sum distinguishing if f(u)≠f(v) for each edge uv∈E(G), where f(v)=∑wv∈E(G)c(wv)+c(v). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χ″∑(G). It has been conjectured that χ″∑(G)≤∆(G)+3 for any simple graph G. Let mad(G)=max{2|E(H)||V(H)|:H⊆G} be the maximum average degree of G. In this paper, by using the famous Combinatorial Nullstellensatz, we prove χ″∑(G)≤max{9,∆(G)+2} for any graph G with mad(G)<4. Furthermore, we characterize the neighbor sum distinguishing total chromatic number for every graph G with mad(G)<4 and ∆(G)≥8.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v56n2.23.06
Journal of Mathematical Study, Vol. 56 (2023), Iss. 2 : pp. 206–218
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Neighbor sum distinguishing total coloring combinatorial nullstellensatz maximum average degree.
Author Details
Danjun Huang Email
Dan Bao Email