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Forbidden Subgraphs for a Graph to Have a Hamiltonian Path Square

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Abstract

The square of a graph is obtained by adding an edge between each pair of vertices that are of distance 2 in the original graph. If a graph G has a hamiltonian path (or cycle) square, then the square of the hamiltonian path (or cycle) is called a hamiltonian path (or cycle) square of G. Chen and Shan (Graphs Comb 31:2113–2124, 2015) characterized all forbidden pairs for a 4-connected graph to have a hamiltonian cycle square. In this paper, we completely characterize pairs of connected forbidden subgraphs for a k-connected (\(k\in \{1,2,3,4\}\)) graph with maximal degree at least 4 to have a hamiltonian path square. Note that the maximal degree condition is necessary for graphs of order at least 5 to have a hamiltonian path square.

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Acknowledgements

The author would like to thank anonymous referees for careful reading and helpful comments. This work is supported by Natural Science Funds of China (Nos. 11871099, 11671037) and by Nature Science Foundation of Qinghai Province (No. 2018-ZJ-717).

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Authors and Affiliations

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

    Xiaojing Yang & Liming Xiong

  2. School of Mathematics and Satistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Xiaojing Yang

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  1. Xiaojing Yang
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  2. Liming Xiong
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Correspondence to Liming Xiong.

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Yang, X., Xiong, L. Forbidden Subgraphs for a Graph to Have a Hamiltonian Path Square. Graphs and Combinatorics 36, 1445–1456 (2020). https://doi.org/10.1007/s00373-020-02186-5

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  • DOI: https://doi.org/10.1007/s00373-020-02186-5

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