Improper coloring of graphs with no odd clique minor

TL;DR

This paper investigates the structure of graphs without odd clique minors, proving weaker variants of a conjecture relating to their colorability and providing improved partition bounds.

Contribution

It introduces two new theorems that partition such graphs into sets with bounded degree or component size, improving previous bounds.

Findings

Graphs with no odd $K_t$ minor can be partitioned into sets with bounded maximum degree.

Graphs with no odd $K_t$ minor can be partitioned into sets with components of bounded size.

The bounds on the number of sets are significantly improved over previous results.

Abstract

As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $6t-9$ sets $V_1, \dots, V_{6t-9}$ such that each $V_i$ induces a subgraph of bounded maximum degree. Secondly, we prove that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $10t-13$ sets $V_1, \dots, V_{10t-13}$ such that each $V_i$ induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into $496t$ such sets.