Toughness, Forbidden Subgraphs and Pancyclicity

Wei Zheng, Hajo Broersma*, Ligong Wang

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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Abstract

Motivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of K1∪ P4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph K1∪ P4 itself. The hamiltonicity of 1-tough K1∪ P4-free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough K1∪ P4-free graphs we give infinite families of graphs that are not pancyclic.

Original languageEnglish
Pages (from-to)839-866
JournalGraphs and combinatorics
Volume37
Early online date19 Feb 2021
DOIs
Publication statusPublished - May 2021

Keywords

  • Forbidden subgraph
  • Hamiltonian graph
  • Pancyclic graph
  • Toughness
  • UT-Hybrid-D

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