Cycle Extendability of Hamiltonian Strongly Chordal Graphs

Abstract

In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle extendable. After a series of papers confirming the conjecture for a number of graph classes, the conjecture is yet refuted by Lafond and Seamone in 2015. Given that their counterexamples are not strongly chordal graphs and they are all only 2-connected, Lafond and Seamone asked the following two questions:(1) Are Hamiltonian strongly chordal graphs cycle extendable? (2) Is there an integer k such that all k-connected Hamiltonian chordal graphs are cycle extendable? Later, a conjecture stronger than Hendry's is proposed. In this paper, we resolve all these questions in the negative. On the positive side, we add to the list of cycle extendable graphs two more graph classes, namely, Hamiltonian 4-leaf powers and Hamiltonian {4-FAN, A}-free chordal graphs.

Further results on Hendry's Conjecture

“…In Section 3, we examine ways in which we can modify the construction given in Section 2 to obtain examples which satisfy even stronger conditions in terms of induced path lengths (improving on a result from [10]), connectivity (answering a question from [10], S-extendibility (answering a conjecture from [3]), and underlying tree structure (showing that results in [1] are almost best possible). Note that similar results to some of those presented in Sections 2 and 3 were independently obtained by Rong et al [13]. Finally, we propose an extremal problem in Section 4 related to our counterexample constructions.…”