On Edge Coloring of Multigraphs
Guangming Jing

TL;DR
This paper introduces a polynomial-time algorithm for edge coloring multigraphs within a bound that confirms a longstanding conjecture, nearly optimal for computational efficiency, and advances the understanding of graph coloring complexity.
Contribution
The paper provides the first polynomial-time algorithm that colors multigraphs with at most 1 more color than the conjectured optimal bound, confirming the conjecture algorithmically.
Findings
Algorithm confirms the Goldberg-Seymour conjecture for multigraphs.
Colors multigraphs within 1 of the optimal chromatic index.
Establishes the computational complexity boundary for edge coloring algorithms.
Abstract
Let and be the maximum degree and chromatic index of a graph , respectively. Appearing in different forms, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) made the following conjecture: Every multigraph satisfies , where is the density of . In this paper, we present a polynomial-time algorithm for coloring any multigraph with colors, confirming the conjecture algorithmically. Since , this algorithm gives a proper edge coloring that uses at most one more color than the optimum. As determining the chromatic index of an arbitrary graph is -hard, the $\max\{ \Delta(G) + 1,…
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Advanced Graph Theory Research · Limits and Structures in Graph Theory
