Chorded pancyclic properties in claw-free graphs

Abstract

A graph G is (doubly) chorded pancyclic if G contains a (doubly) chorded cycle of every possible length m for 4 ≤ m ≤ |V (G)|. In 2018, Cream, Gould, and Larsen completely characterized the pairs of forbidden subgraphs that guarantee chorded pancyclicity in 2-connected graphs. In this paper, we show that the same pairs also imply doubly chorded pancyclicity. We further characterize conditions for the stronger property of doubly chorded (k, m)-pancyclicity where, for k ≤ m ≤ |V (G)|, every set of k vertices in G is contained in a doubly chorded i-cycle for all m ≤ i ≤ |V (G)|. In particular, we examine forbidden pairs and degree sum conditions that guarantee this recently defined cycle property.