Odd-Sum Colorings of Planar Graphs
Daniel W. Cranston

TL;DR
This paper investigates odd-sum colorings of planar graphs, providing negative answers to previous open questions about bounds related to girth and maximum degree, and refutes a related conjecture.
Contribution
The authors disprove the existence of certain bounds for odd-sum colorings in planar graphs with specified girth and degree, addressing open problems and conjectures in the field.
Findings
Both questions about bounds for odd-sum colorings are answered negatively.
A conjecture by Caro et al. is refuted.
Progress is made on an additional open problem.
Abstract
A \emph{coloring} of a graph is a map such that for all . A coloring is an \emph{odd-sum} coloring if is odd, for each vertex . The \emph{odd-sum chromatic number} of a graph , denoted , is the minimum number of colors used (that is, the minimum size of the range) in an odd-sum coloring of . Caro, Petru\v{s}evski, and \v{S}krekovski showed, among other results, that is well-defined for every finite graph and, in fact, . Thus, for every planar graph (by the 4 Color Theorem), for every triangle-free planar graph (by Gr\"{o}tzsch's Theorem), and for every bipartite graph. Caro et al. asked, for every even , whether there exists such that…
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Taxonomy
TopicsAdvanced Graph Theory Research
