Edge coloring of graphs of signed class 1 and 2
Robert Janczewski, Krzysztof Turowski, Bart{\l}omiej Wr\'oblewski

TL;DR
This paper introduces new classes of graphs based on the chromatic index of signed graphs, showing which graphs have a signature-independent chromatic index and characterizing their properties.
Contribution
It defines classes $1^\ ext{\pm}$ and $2^\pm$ of graphs where the chromatic index is signature-independent, and characterizes these classes with examples and necessary conditions.
Findings
All wheels, necklaces, and certain bipartite graphs are of class $1^\pm$.
Graphs of class $2^\pm$ must have odd maximum degree.
Provides conditions for a graph to be of class $2^\pm$.
Abstract
Recently, Behr introduced a notion of the chromatic index of signed graphs and proved that for every signed graph , it holds that \[ \Delta(G)\leq\chi'(G\text{, }\sigma)\leq\Delta(G)+1\text{,} \] where is the maximum degree of and denotes its chromatic index. In general, the chromatic index of , depends on both the underlying graph and the signature . In the paper we study graphs for which , does not depend on . To this aim we introduce two new classes of graphs, namely and , such that graph is of class (respectively, ) if and only if , (respectively, , ) for all possible signatures . We prove that all wheels, necklaces, complete bipartite graphs with and almost all…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems
