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Hamilton Cycles in Implicit 2-Heavy Graphs

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Abstract

Let id(v) denote the implicit degree of a vertex v in a graph G. We define G of order n to be implicit 2-heavy if at least two of the end vertices of each induced claw have implicit degree at least \(\frac{n}{2}\). In this paper, we show that every implicit 2-heavy graph G is hamiltonian if we impose certain additional conditions on the connectivity of G or forbidden induced subgraphs. Our results extend two previous theorems of Broersma et al. (Discret Math 167–168:155–166, 1997) on the existence of Hamilton cycles in 2-heavy graphs.

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Acknowledgments

The authors are very grateful to the anonymous referee whose helpful comments and suggestions have led to a substantially improvement of the paper.

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Authors and Affiliations

  1. School of Management, Qufu Normal University, Rizhao, 276826, People’s Republic of China

    Junqing Cai

  2. L.R.I, UMR 8623, CNRS, Université Paris-Sud 11, 91405, Orsay, France

    Hao Li

  3. Institute for Interdisciplinary Research, Jianghan University, Wuhan, 430019, People’s Republic of China

    Hao Li

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  1. Junqing Cai
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  2. Hao Li
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Correspondence to Junqing Cai.

Additional information

This work is supported by Postdoctoral Science Foundation of China (No. 2015M571999), National Natural Science Foundation of China (Nos. 11501322, 11426145), Natural Science Foundation of Shandong Province (No. ZR2014AP002) and Scientific Research Foundation for Doctors in Qufu Normal University (No. 2012015).

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Cai, J., Li, H. Hamilton Cycles in Implicit 2-Heavy Graphs. Graphs and Combinatorics 32, 1329–1337 (2016). https://doi.org/10.1007/s00373-015-1669-4

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