Longest cycles in vertex-transitive and highly connected graphs

TL;DR

This work advances two central questions about longest paths and cycles in graphs by establishing new asymptotic bounds: in connected vertex-transitive graphs on $n$ vertices there exists a cycle of length at least $Ω(n^{13/21})$, and in every $r$-connected graph with $r\ge2$, any two longest cycles meet in at least $Ω(r^{5/8})$ vertices. The core technique is a local separator lemma bounding the size of a separator between two longest cycles by $|S|\le ex(2k,\{Q_3^+,K_{3,3}\})=O(k^{8/5})$, which, combined with a reduction to a computer-assisted linear-programming framework on configuration graphs, yields the bounds. The proofs synthesize combinatorial arguments, computer search, and LP methods to push beyond the previous $Ω(n^{3/5})$ and $Ω(r^{3/5})$ bounds, offering a pathway toward resolving Lovász’s and Thomassen’s conjectures and Smith’s intersection conjecture. The approach also identifies structural limitations and suggests avenues for further improvements via larger extremal graphs and refined configurations.

Abstract

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász 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 $r\ge 2$ 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 $n\geq 3$ 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 $r\geq 2$, 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.

Figures (3)

• Figure 1: The three ordered bipartite graphs $(H,\sigma_X,\sigma_Y)$ that need to be considered up to equivalence, as defined in \ref{['lem:weightlongcycle']}. The permutations $\sigma_X,\sigma_Y$ are implicit in the drawings.
• Figure 2: The above depicts a bipartite graph $H$ (left) together with a possible configuration graph $G_\tau$ (right). The coloured edges are safe.
• Figure 3: Example of a weighted configuration graph on an ordered bipartite graph with $H=Q_3$, for which there does not exist any good long cycle. The weight edges are specified as edge labels. Dangerous edges are dashed.

Theorems & Definitions (16)

• Theorem 1.1
• Conjecture 1.2: cf. Conjecture 5.2 in Groetschel84 and Conjecture 4.15 in Bondy95
• Theorem 1.3
• Lemma 1.4
• proof : Proof of Theorem \ref{['thm:rconnected_corollary']}
• proof : Proof of Theorem \ref{['thm:long_cycle']}
• Lemma 3.1
• proof
• proof
• Lemma 3.3