A counterexample to Hickingbotham's conjecture about $k$-ghost-edges

TL;DR

The paper investigates whether Hickingbotham's conjecture on $k$-ghost-edges holds for graphs with $tw(G)\le k$. It introduces $k$-ghost-edges and demonstrates a counterexample: a graph with $tw(G)=4$ and four internally vertex-disjoint $(x,y)$-paths for which $xy$ is a $4$-ghost-edge, contradicting the conjecture. The authors achieve this via an explicit graph construction and a detailed analysis of tree decompositions of width $4$, establishing that $xy$ must appear in some bag in every such decomposition. This refutes the proposed threshold behavior and refines understanding of how disjoint paths interact with ghost-edge properties under tree decompositions, with implications for related conjectures in graph minors and width parameters.

Abstract

Fix $k\in \mathbb{N}$ and let $G$ be a connected graph with $tw(G)\leq k$. We say that $xy\in E(G^c)$ is a {\em $k$-ghost-edge} of $G$ if for every tree decomposition $(T,\cB)$ of $G$ with width at most $k$, the set $\{x,y\}$ is contained in a bag of $(T,\cB)$. Although a $k$-ghost-edge of $G$ is not an edge of $G$, but it behaves like real edges with respect to tree decomposition of $G$ with width at most $k$. For any graph $G$ with treewidth $k$ and $xy\in E(G^c)$, when there are at least $k+1$ internally vertex disjoint $(x,y)$-paths, Hickingbotham proved that $xy$ is a $k$-ghost-edge of $G$; while when there are at most $k$ internally vertex disjoint $(x,y)$-paths, he conjectured that it is not a $k$-ghost-edge of $G$. In this paper, we prove that this conjecture is wrong.

Figures (2)

• Figure 1: The bold connected subgraphs in $G$ are branch sets of a $K_5$-minor of $G$.
• Figure 2: a tree decomposition of $H_1$ with width $4$ and with $\{x,y,d_2\}$ contained in a bag.

Theorems & Definitions (12)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Lemma 2.1
• Theorem 2.2
• proof
• claim 2.2.1
• proof
• claim 2.2.2