Small hitting sets for longest paths and cycles

TL;DR

This paper improves bounds on the minimum size of vertex sets hitting all longest paths and cycles in graphs, and applies these results to advance conjectures on path and cycle lengths in vertex-transitive graphs.

Contribution

The paper introduces tighter bounds for hitting sets of longest paths and cycles, and connects these bounds to progress on longstanding conjectures in graph theory.

Findings

Connected graphs have hitting sets of size at most √8n for longest paths and cycles.

Vertex-transitive graphs contain long paths and cycles of length proportional to n^{9/14}.

New bounds improve previous results from O(n^{2/3}) to √8n and related exponents.

Abstract

Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the well-known conjectures of Lov\'{a}sz (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertex-transitive graphs, we present improved bounds for the parameters $\mathrm{lpt}(G)$ and $\mathrm{lct}(G)$, defined as the minimum size of a set of vertices in a graph $G$ hitting all longest paths (cycles, respectively). First, we show that every connected graph $G$ on $n$ vertices satisfies $\mathrm{lpt}(G)\le \sqrt{8n}$, and $\mathrm{lct}(G)\le \sqrt{8n}$ if $G$ is additionally $2$-connected. This improves a sequence of earlier bounds for these problems, with the previous state of the art being $O(n^{2/3})$. Second, we show that every connected graph $G$ satisfies $\mathrm{lpt}(G)\le O(\ell^{5/9})$, where $\ell$ denotes the maximum length of a path in $G$. As an immediate application of this latter bound, we present further progress towards Lov\'{a}sz' and Thomassen's conjectures: We show that every connected vertex-transitive graph of order $n$ contains a cycle (and path) of length $\Omega(n^{9/14})$. This improves the previous best bound of the form $\Omega(n^{13/21})$. Interestingly, our proofs make use of several concepts and results from structural graph theory, such as a result of Robertson and Seymour (1990) on transactions in societies and Tutte's $2$-separator theorem.