Let G be a graph, \(\nu \) the order of G and k a positive integer such that \(k\le (\nu -2)/2\). Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A graph G is Hamiltonian if it contains a Hamiltonian cycle. A graph G is Hamiltonian-connected if, for any two of its vertices, it contains a spanning path joining the two vertices. In this paper, we discuss k-extendable nonbipartite graphs with \(\kappa (G)\ge 2k+r\) where \(k\ge 1\) and \(r\ge 0\). It is shown that if \(\nu \le 6k+2r\), then G is Hamiltonian; and if \(\nu > 6k+2r\), then G has a longest cycle C such that \(|V(C)|\ge 6k+2r\); and if \(\nu <6k+2r\), then G is Hamiltonian-connected; and if \(\nu \ge 6k+2r\), then for each pair of vertices \(z_1\) and \(z_2\) of G, there is a path P of G joining \(z_1\) and \(z_2\) such that \(|V(P)|\ge 6k+2r-2\). All the bounds are sharp and all results can be extended to 2k-factor-critical graphs.
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The authors are grateful to the referees for their careful reading and constructive suggestions that improved the quality of the paper greatly.
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Yanping Xu is co-first author.
Zhiyong Gan was supported by the State Scholarship Fund awarded by the China Scholarship Council (Grant number 201906755002), and the Scholarship of South China Normal University for Studying Abroad.
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Gan, Z., Lou, D. & Xu, Y. Hamiltonian Cycle Properties in k-Extendable Non-bipartite Graphs with High Connectivity. Graphs and Combinatorics 36, 1043–1058 (2020). https://doi.org/10.1007/s00373-020-02164-x
- Nonbipartite graph
- Long cycle
Mathematics Subject Classification