Odd complete bipartite minors in graphs with independence number two

TL;DR

The paper addresses the problem of obtaining odd $K_{\ell,\chi(G)-\ell}$-minors in graphs with independence number at most two, extending the known minor version of this result to the odd-minor setting. The authors develop the notion of a special odd model and prove a main lemma asserting the existence of such a model for any $2\ell \le \lceil n/2\rceil$ in an $n$-vertex graph with $\alpha(G)=2$, via a structural argument blending Blasiak-type connectivity, induced $P_3$-path packings, and critical-graph analysis. This leads to the conclusion that every $\alpha(G)\le 2$ graph contains, as an odd minor, $K_{\ell,\chi(G)-\ell}$ for all $1\le \ell<\chi(G)$—a strengthening of the Odd Hadwiger conjecture for this graph class. The results contribute to the broader Hadwiger-program under restricted independence and provide tools for potential extensions to wider graph classes.

Abstract

Recently, Chen and Deng have proved that every graph $G$ with independence number two contains $K_{\ell,χ(G)- \ell}$ as a minor for each integer $\ell$ with $1\leq\ell < χ(G)$. In this paper, we extend this result to odd minor version. That is, we prove that each graph $G$ with independence number two contains $K_{\ell,χ(G)- \ell}$ as an odd minor for each integer $\ell$ with $1\leq\ell < χ(G)$.

Theorems & Definitions (22)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Conjecture 1.4
• Theorem 1.5
• Theorem 1.6
• Conjecture 1.7
• Lemma 2.1
• proof
• Theorem 2.2