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Pairs of disjoint matchings and related classes of graphs

Huizheng Guo, Kieran Kaempen, Zhengda Mo, Sam Qunell, Joe Rogge, Chao Song, Anush Tserunyan and Jenna Zomback

Vol. 16 (2023), No. 2, 249–264

For a finite graph G, we study the maximum 2-edge colorable subgraph problem and a related ratio μ(G)ν(G), where ν(G) is the matching number of G, and μ(G) is the size of the largest matching in any pair (H,H) of disjoint matchings maximizing |H| + |H| (equivalently, forming a maximum 2-edge colorable subgraph). Previously, it was shown that 4 5 μ(G)ν(G) 1, and the class of graphs achieving 4 5 was completely characterized. We show here that any rational number between 4 5 and 1 can be achieved by a connected graph. Furthermore, we prove that every graph with ratio less than 1 must admit special subgraphs.

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graph, matching, edge coloring, maximum 2-edge colorable subgraph
Mathematical Subject Classification
Primary: 05C70
Received: 7 July 2021
Revised: 8 May 2022
Accepted: 8 May 2022
Published: 26 May 2023

Communicated by Ronald Gould
Huizheng Guo
Department of Mathematics
The George Washington University
Washington, DC
United States
Kieran Kaempen
Department of Computer Science
University of Illinois Urbana-Champaign
Urbana, IL
United States
Zhengda Mo
Department of Mathematics
University of Illinois Urbana-Champaign
Urbana, IL
United States
Sam Qunell
Department of Mathematics
University of California
Los Angeles, CA
United States
Joe Rogge
Department of Mathematics
University of Washington
Seattle, WA
United States
Chao Song
Department of Mathematics
University of Illinois Urbana-Champaign
Urbana, IL
United States
Anush Tserunyan
Mathematics and Statistics Department
McGill University
Montreal, QC
Jenna Zomback
Department of Mathematics
University of Illinois Urbana-Champaign
Urbana, IL
United States