BOOK-CHAPTER

Edge-Coloring Algorithms for Bounded Degree Multigraphs

Abhishek Dhawan

Year: 2024 Society for Industrial and Applied Mathematics eBooks Pages: 2120-2157   Publisher: Society for Industrial and Applied Mathematics

Abstract

In this paper, we consider algorithms for edge-coloring multigraphs G of bounded maximum degree, i.e., Δ (G) = O(1). Shannon's theorem states that any multigraph of maximum degree Δ can be properly edge- colored with ⌊3Δ/2⌋ colors. Our main results include algorithms for computing such colorings. We design deterministic and randomized sequential algorithms with running time O(n log n) and O(n), respectively. This is the first improvement since the O(n2) algorithm in Shannon's original paper, and our randomized algorithm is optimal up to constant factors. We also develop distributed algorithms in the LOCAL model of computation. Namely, we design deterministic and randomized LOCAL algorithms with running time Õ(log5 n) and O(log2 n), respectively. The deterministic sequential algorithm is a simplified extension of earlier work of Gabow et al. in edge-coloring simple graphs. The other algorithms apply the entropy compression method in a similar way to recent work by the author and Bernshteyn, where the authors design algorithms for Vizing's theorem for simple graphs. We also extend those results to Vizing's theorem for multigraphs.

Keywords:
Multigraph Deterministic algorithm Degree (music) Bounded function Randomized algorithm Mathematics Edge coloring Combinatorics Greedy coloring Algorithm Discrete mathematics Freivalds' algorithm Graph Line graph

Metrics

2
Cited By
4.63
FWCI (Field Weighted Citation Impact)
0
Refs
0.88
Citation Normalized Percentile
Is in top 1%
Is in top 10%

Citation History

Topics

Complexity and Algorithms in Graphs
Physical Sciences →  Computer Science →  Computational Theory and Mathematics
Machine Learning and Algorithms
Physical Sciences →  Computer Science →  Artificial Intelligence
Computability, Logic, AI Algorithms
Physical Sciences →  Computer Science →  Computational Theory and Mathematics

Related Documents

JOURNAL ARTICLE

Mixed hypergraphs with bounded degree: edge-coloring of mixed multigraphs

Daniel Král͏̌Jan Kratochvı́lHeinz‐Jürgen Voss

Journal:   Theoretical Computer Science Year: 2003 Vol: 295 (1-3)Pages: 263-278
JOURNAL ARTICLE

Algorithms for Edge Coloring Bipartite Graphs and Multigraphs

Harold N. GabowOded Kariv

Journal:   SIAM Journal on Computing Year: 1982 Vol: 11 (1)Pages: 117-129
JOURNAL ARTICLE

On-Line Edge-Coloring Algorithms for Degree-Bounded Bipartite Graphs

Masakuni TakiMikihito SugiuraToshinobu Kashiwabara

Journal:   IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences Year: 2002 Vol: 85 (5)Pages: 1062-1065
JOURNAL ARTICLE

Edge-coloring of multigraphs

Martin KocholNad’a KrivoňákováSilvia Smejová

Journal:   Discrete Mathematics Year: 2005 Vol: 300 (1-3)Pages: 229-234
JOURNAL ARTICLE

Edge-coloring almost bipartite multigraphs

Tomás FederCarlos Subi

Journal:   Information Processing Letters Year: 2013 Vol: 113 (18)Pages: 685-689
© 2026 ScienceGate Book Chapters — All rights reserved.