Totally odd immersions of complete graphs in graph products

TL;DR

This paper investigates the maximum size of totally odd immersions of complete graphs within various graph products, extending previous work on standard immersions and providing insights related to the Odd Hadwiger Conjecture.

Contribution

It introduces the parameter $toi(G)$ for totally odd immersions and analyzes its behavior under different graph products, showing limitations on constructing counterexamples to the odd immersion conjecture.

Findings

No minimal counterexample to the odd immersion conjecture can be formed from certain graph products.

The parameter $toi(G)$ behaves differently from standard immersion parameters under graph products.

The results restrict possible constructions of counterexamples to the Odd Hadwiger Conjecture.

Abstract

For a graph $G$, let $im(G)$ denote the maximum integer $t$ such that $G$ contains $K_t$ as an immersion. A recent paper of Collins, Heenehan, and McDonald (2023) studied the behaviour of this parameter under graph products, asking how large can $im(G\ast H)$ be in terms of $im(G)$ and $im(H)$, when $\ast$ is one of the four standard graph products. We consider a similar question for the parameter $toi(G)$ which denotes the maximum integer $t$ such that $G$ contains $K_t$ as a totally odd immersion. As an application, we obtain that no minimum counterexample to the immersion-analogue of the Odd Hadwiger Conjecture can be obtained from the Cartesian, direct (tensor), lexicographic or strong product of graphs.