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Strongly Spanning Trailable Graphs with Small Circumference and Hamilton-Connected Claw-Free Graphs

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Abstract

A graph G is strongly spanning trailable if for any \(e_1=u_1v_1, e_2=u_2v_2\in E(G)\) (possibly \(e_1=e_2\)), \(G(e_1, e_2)\), which is obtained from G by replacing \(e_1\) by a path \(u_1v_{e_1}v_1\) and by replacing \(e_2\) by a path \(u_2v_{e_2}v_2\), has a spanning \((v_{e_1}, v_{e_2})\)-trail. A graph G is Hamilton-connected if there is a spanning path between any two vertices of V(G). In this paper, we first show that every 2-connected 3-edge-connected graph with circumference at most 8 is strongly spanning trailable with an exception of order 8. As applications, we prove that every 3-connected \(\{K_{1, 3}, N_{1, 2, 4}\}\)-free graph is Hamilton-connected and every 3-connected \(\{K_{1, 3}, P_{10}\}\)-free graph is Hamilton-connected with a well-defined exception. The last two results extend the results in Hu and Zhang (Graphs Comb 32: 685–705, 2016) and Bian et al. (Graphs Comb 30: 1099–1122, 2014) respectively.

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Acknowledgements

The authors thank the referees very much for their carefully reading. The work is supported by the Natural Science Funds of China (nos: 11871099 and 11671037).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Xia Liu

  2. School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Liming Xiong

  3. Department of Mathematics, West Virginia University, Morgantown, WV, 26506, USA

    Hong-Jian Lai

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  1. Xia Liu
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  2. Liming Xiong
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  3. Hong-Jian Lai
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Correspondence to Liming Xiong.

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Liu, X., Xiong, L. & Lai, HJ. Strongly Spanning Trailable Graphs with Small Circumference and Hamilton-Connected Claw-Free Graphs. Graphs and Combinatorics 37, 65–85 (2021). https://doi.org/10.1007/s00373-020-02236-y

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