Discussiones Mathematicae Graph Theory (Feb 2020)

A Note on Cycles in Locally Hamiltonian and Locally Hamilton-Connected Graphs

  • Tang Long,
  • Vumar Elkin

DOI
https://doi.org/10.7151/dmgt.2124
Journal volume & issue
Vol. 40, no. 1
pp. 77 – 84

Abstract

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Let 𝒫 be a property of a graph. A graph G is said to be locally 𝒫, if the subgraph induced by the open neighbourhood of every vertex in G has property 𝒫. RyjÑček conjectures that every connected, locally connected graph is weakly pancyclic. Motivated by the above conjecture, van Aardt et al. [S.A.van Aardt, M. Frick, O.R. Oellermann and J.P.de Wet, Global cycle properties in locally connected, locally traceable and locally Hamiltonian graphs, Discrete Appl. Math. 205 (2016) 171–179] investigated the global cycle structures in connected, locally traceable/Hamiltonian graphs. Among other results, they proved that a connected, locally Hamiltonian graph G with maximum degree at least |V (G)| βˆ’ 5 is weakly pancyclic. In this note, we improve this result by showing that such a graph with maximum degree at least |V (G)|βˆ’6 is weakly pancyclic. Furthermore, we show that a connected, locally Hamilton-connected graph with maximum degree at most 7 is fully cycle extendable.

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