Characterizing Forbidden Subgraphs That Imply Pancyclicity In 4-Connected, Claw-Free Graphs

In 1984, Matthews and Sumner conjectured that every 4-connected, claw-free graph contains a Hamiltonian cycle. This still unresolved conjecture has been the motivation for research into the existence of other cycle structures. In this paper, we consider the stronger property of pancyclicity for 4-connected graphs. In particular, we show that every 4-connected, {K-1,K-3 , N(i , j , k)}-free graph, where i , j , k >= 1 and i + j + k = 6, is pancyclic. This, together with results by Ferrara, Morris, Wenger, and Ferrara et al. completes a characterization of the graphs Y such that every {K-1,K-3, Y}-free graph is pancyclic. In addition, this represents the best known progress towards answering a question of Gould concerning a characterization of the pairs of forbidden subgraphs that imply pancyclicity in 4-connected graphs. (C) 2021 Elsevier B.V. All rights reserved.