Abstract
An edge e in a matching covered graph G is removable if \(G-e\) is matching covered. Removable edges were introduced by Lovász and Plummer in connection with ear decompositions of matching covered graphs. A brick is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Lovász proved that every brick other than \(K_4\) and \(\overline{C_6}\) has a removable edge. It is known that every 3-connected claw-free graph with even number of vertices is a brick. By characterizing the structure of adjacent non-removable edges, we show that every claw-free brick G with more than 6 vertices has at least 5|V(G)|/8 removable edges.
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Acknowledgements
The authors would like to thank the anonymous referees for helpful comments on improving the representation of the paper.
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The research is partially supported by NSFC (Nos. 12271235, 12361070, 12371355), NSF of Fujian Province (Nos. 2021J06029, 2023J01909) and Fujian Key Laboratory of Granular Computing and Applications (Minnan Normal University).
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Wu, X., Lu, F. & Zhang, L. Removable Edges in Claw-Free Bricks. Graphs and Combinatorics 40, 43 (2024). https://doi.org/10.1007/s00373-024-02769-6
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DOI: https://doi.org/10.1007/s00373-024-02769-6