Odd clique minors in graphs with independence number two

TL;DR

This work investigates Odd Hadwiger's Conjecture for graphs with independence number $\alpha(G)\le 2$, establishing that in this class an odd $K_t$-minor exists whenever an odd clique minor of order $t=\chi(G)$ is forced. A key contribution is showing that $oh(G)\ge \chi(G)$ is equivalent to $oh(G)\ge \lceil n/2\rceil$ for $\alpha(G)\le 2$, and proving a main theorem that guarantees this bound when $G$ is $H$-free for certain induced subgraphs $H$ with $\alpha(H)\le 2$ (each $H$ induced in one of $K_1+P_4$, $K_2+(K_1\cup K_3)$, $K_1+(K_1\cup K_4)$, $K_7^-$, $K_7$, or the kite). The proof proceeds by contradiction on a minimum counterexample and conducts a detailed five-case analysis tied to the forbidden subgraphs, employing structural tools such as seagulls and inflations of $C_5$ to construct odd clique minors of order $\lceil n/2\rceil$. These results contribute new evidence towards Odd Hadwiger's Conjecture in the $\alpha(G)=2$ regime and sharpen understanding of how induced-subgraph restrictions influence odd-minor behavior. The findings also situate the conjecture within a broader landscape of Hadwiger-type results for specialized graph classes and suggest pathways for extending the approach to additional forbidden-subgraph families and related graph classes.

Abstract

A $K_t$-expansion consists of $t$ vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-colored so that the edges of the trees are bichromatic but the edges between trees are monochromatic. A graph contains an odd $K_t$ minor or an odd clique minor of order $t$ if it contains an odd $K_t$-expansion. Gerards and Seymour from 1995 conjectured that every graph $G$ contains an odd $K_{χ(G)}$ minor, where $χ(G)$ denotes the chromatic number of $G$. This conjecture is referred to as ``Odd Hadwiger's Conjecture". Let $α(G)$ denote the independence number of a graph $G$. In this paper we investigate the Odd Hadwiger's Conjecture for graphs $G$ with $α(G)\le2$. We first observe that a graph $G$ on $n$ vertices with $α(G)\le2$ contains an odd $K_{χ(G)}$ minor if and only if $G$ contains an odd clique minor of order $\lceil n/2\rceil$. We then prove that every graph $G$ on $n$ vertices with $α(G)\le 2$ contains an odd clique minor of order $\lceil n/2\rceil$ if $G$ contains a clique of order $n/4$ when $n$ is even and $(n+3)/4$ when $n$ is odd, or $G$ does not contain $H$ as an induced subgraph, where $α(H)\le 2$ and $H$ is an induced subgraph of $K_1 + P_4$, $K_2+(K_1\cup K_3)$, $K_1+(K_1\cup K_4)$, $K_7^-$, $K_7$, or the kite graph.

Theorems & Definitions (26)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3: Chudnovsky and Seymour cs
• Theorem 1.4
• proof
• Theorem 1.5: Chudnovsky and Seymour cs
• Lemma 1.6: Chen and Deng cd
• Theorem 1.7
• proof
• Theorem 1.8: Song and B. Thomas st