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Sublinear Longest Path Transversals

James A. Long, Kevin G. Milans, Andrea Munaro

TL;DR

The paper investigates sublinear long-path and long-cycle transversals in graphs by introducing a unified framework based on $R$-subdivisions. The authors prove a general bound: if the maximum $R$-subdivisions in a graph $G$ pairwise intersect, then the $R$-transversal number satisfies $\tau_R(G) \le 8 m^{5/4} n^{3/4}$, using an $\varepsilon$-partial transversal method together with separator/connector arguments grounded in Menger's theorem. This leads to concrete corollaries: for connected graphs, $lpt(G) \le 8 n^{3/4}$, and for $2$-connected graphs, $lct(G) \le 20 n^{3/4}$, advancing sublinear bounds toward the longstanding questions about constant-size transversals. The paper also shows that high connectivity, $\kappa(G) > m^2$, ensures pairwise intersection of maximum $R$-subdivisions, thereby enabling the sublinear transversal bound in broader cases. Overall, the work provides a general, connectivity-sensitive approach to small transversals for families defined by maximum subdivisions, with immediate implications for longest path and longest cycle transversals.

Abstract

We show that connected graphs admit sublinear longest path transversals. This improves an earlier result of Rautenbach and Sereni and is related to the fifty-year-old question of whether connected graphs admit longest path transversals of constant size. The same technique allows us to show that $2$-connected graphs admit sublinear longest cycle transversals.

Sublinear Longest Path Transversals

TL;DR

The paper investigates sublinear long-path and long-cycle transversals in graphs by introducing a unified framework based on -subdivisions. The authors prove a general bound: if the maximum -subdivisions in a graph pairwise intersect, then the -transversal number satisfies , using an -partial transversal method together with separator/connector arguments grounded in Menger's theorem. This leads to concrete corollaries: for connected graphs, , and for -connected graphs, , advancing sublinear bounds toward the longstanding questions about constant-size transversals. The paper also shows that high connectivity, , ensures pairwise intersection of maximum -subdivisions, thereby enabling the sublinear transversal bound in broader cases. Overall, the work provides a general, connectivity-sensitive approach to small transversals for families defined by maximum subdivisions, with immediate implications for longest path and longest cycle transversals.

Abstract

We show that connected graphs admit sublinear longest path transversals. This improves an earlier result of Rautenbach and Sereni and is related to the fifty-year-old question of whether connected graphs admit longest path transversals of constant size. The same technique allows us to show that -connected graphs admit sublinear longest cycle transversals.

Paper Structure

This paper contains 2 sections, 4 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Maximum subdivision transversals

Key Result

Theorem 1

Let $R$ be a connected $m$-edge multigraph with $m\ge 1$ and let $G$ be an $n$-vertex graph. If the maximum $R$-subdivisions in $G$ pairwise intersect, then $\tau_R(G) \le 8m^{5/4}n^{3/4}$.

Figures (2)

  • Figure 1: The graph $G_0$: a $12$-vertex graph with $\mathrm{lpt}(G_0)=2$.
  • Figure 2: $(V(C),V(F))$-connector case. The subpath $W$ of the cycle $C$ is dashed, and the cycle $D$ is displayed in bold.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Lemma 3
  • proof
  • Corollary 4