Total colorings of planar graphs without adjacent triangles
type: Item
instance of: scholarly article
title: Total colorings of planar graphs without adjacent triangles
zbMATH Open document ID: 1189.05065
author: Jianfeng Hou
author: Xiang Yong Sun
author: Jian-Liang Wu
author: Yu-Wen Wu
published in: Discrete Mathematics
publication date: 2009-01-28T00:00:00Z
review text: Let \(G=\left(V(G), E(G)\right)\) be a graph with maximum degree \(\Delta\). A total \(k\)-coloring of \(G\) is a coloring of \(V(G) \cup E(G)\) using \(k\) colors such that no two adjacent or incident elements receive the same color. The total chromatic number \(\chi''(G)\) is the smallest integer \(k\) such that \(G\) has a total \(k\)-coloring. The well-known Total Coloring Conjecture asserts that \(\Delta+1 \leq \chi''(G) \leq \Delta+2\). A partial result is established in this paper as follows. The Total Coloring Conjecture is true for a planar graph \(G\) without adjacent 3-cycles. Furthermore, \(\chi''(G)=\Delta+1\) if \(\Delta \geq 9\).
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 5499886
zbMATH Keywords: planar graph
zbMATH Keywords: total coloring
zbMATH Keywords: 3-cycle
zbMATH Keywords: total chromatic number
MaRDI profile type: Publication
full work available at URL: j.disc.2007.12.071
OpenAlex ID: W1970135621
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DOI: 10.1016/J.DISC.2007.12.071
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description: scientific article; zbMATH DE number 5499886