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On an induced version of Menger's theorem

Kevin Hendrey, Sergey Norin, Raphael Steiner, Jérémie Turcotte

TL;DR

This work advances induced, distance-separated variants of Menger's theorem by proving that, for $X,Y\subseteq V(G)$ and suitable graph classes, either there are $k$ pairwise non-adjacent $X$-$Y$-paths or a separator of size proportional to $k$. In graphs with bounded maximum degree and in graphs excluding a topological minor, the authors develop a unifying approach based on strong edge colourings, edge contractions, and lifting of paths to enforce non-adjacency. The subcubic results yield concrete, tight bounds: a collection of at least $16k$ disjoint $X$-$Y$-paths with limited external degree guarantees $k$ non-adjacent paths, and for $k=2$ five such paths suffice (computer-verified), with counterexamples clarifying the limits. The paper thus connects induced Menger-type phenomena with graph minor theory, structure theorems, and algorithmic verification, spanning both general theory and concrete, small-degree regimes.

Abstract

We prove Menger-type results in which the obtained paths are pairwise non-adjacent, both for graphs of bounded maximum degree and, more generally, for graphs excluding a topological minor. We further show better bounds in the subcubic case, and in particular obtain a tight result for two paths using a computer-assisted proof.

On an induced version of Menger's theorem

TL;DR

This work advances induced, distance-separated variants of Menger's theorem by proving that, for and suitable graph classes, either there are pairwise non-adjacent --paths or a separator of size proportional to . In graphs with bounded maximum degree and in graphs excluding a topological minor, the authors develop a unifying approach based on strong edge colourings, edge contractions, and lifting of paths to enforce non-adjacency. The subcubic results yield concrete, tight bounds: a collection of at least disjoint --paths with limited external degree guarantees non-adjacent paths, and for five such paths suffice (computer-verified), with counterexamples clarifying the limits. The paper thus connects induced Menger-type phenomena with graph minor theory, structure theorems, and algorithmic verification, spanning both general theory and concrete, small-degree regimes.

Abstract

We prove Menger-type results in which the obtained paths are pairwise non-adjacent, both for graphs of bounded maximum degree and, more generally, for graphs excluding a topological minor. We further show better bounds in the subcubic case, and in particular obtain a tight result for two paths using a computer-assisted proof.
Paper Structure (4 sections, 13 theorems, 8 equations, 4 figures)

This paper contains 4 sections, 13 theorems, 8 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Graphs with bounded maximum degree
  3. Excluding a topological minor
  4. Subcubic graphs

Key Result

Theorem 1.1

If $k\in \mathbb{N}$, $G$ is a graph and $X,Y\subseteq V(G)$, then there exists either

Figures (4)

  • Figure 1: Example requiring four colours for any strong edge colouring of non-horizontal edges.
  • Figure 2: Example of a path system $\mathcal{H}$ and two examples for the $\oplus$ operation. The paths are labelled from 1 to 5 from top to bottom.
  • Figure 3: Counter-example (a) in \ref{['thm:twopathssubcubic']}.
  • Figure 4: Counter-example (b) in \ref{['thm:twopathssubcubic']}.

Theorems & Definitions (34)

  • Theorem 1.1: Menger's theorem menger_zur_1927
  • Conjecture 1.2
  • Theorem 1.3: georgakopoulos_graph_2023albrechtsen_induced_2023
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 2.1
  • ...and 24 more