Papers › Longest cycles in vertex-transitive and highly connected graphs
Longest cycles in vertex-transitive and highly connected graphs
Carla Groenland, Sean Longbrake, Raphael Steiner, Jérémie Turcotte, Liana Yepremyan
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We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for r≥2 in every r-connected graph any two longest cycles intersect in at least r vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on n≥3 vertices contains a cycle (and hence path) of length at least Ω(n^(13/21)), improving on Ω(n^(3/5)) from [DeVos, \emph{arXiv:2302:04255}, 2023]. Second, we show that in every r-connected graph with r≥2, any two longest cycles meet in at least Ω(r^(5/8)) vertices, improving on Ω(r^(3/5)) from [Chen, Faudree and Gould, \emph{J. Combin. Theory, Ser.~ B}, 1998]. Our proof combines combinatorial arguments, computer-search and linear programming.
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