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Precoloring extension of Vizing's Theorem for multigraphs
- Publication Year :
- 2022
-
Abstract
- Let $G$ be a graph with maximum degree $\Delta(G)$ and maximum multiplicity $\mu(G)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $\Delta(G)+\mu(G)$. The distance between two edges $e$ and $f$ in $G$ is the length of a shortest path connecting an endvertex of $e$ and an endvertex of $f$. A distance-$t$ matching is a set of edges having pairwise distance at least $t$. Edwards et al. proposed the following conjecture: For any graph $G$, using the palette $\{1, \dots, \Delta(G)+\mu(G)\}$, any precoloring on a distance-$2$ matching can be extended to a proper edge coloring of $G$. Gir\~{a}o and Kang verified this conjecture for distance-$9$ matchings. In this paper, we improve the required distance from $9$ to $3$ for multigraphs $G$ with $\mu(G) \ge 2$.<br />Comment: 23 pages,4 figures
- Subjects :
- Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2204.01074
- Document Type :
- Working Paper