[Paper Review] A new upper bound on the chromatic number of graphs with no odd $K_t$ minor

Sergey Norin, Zi‐Xia Song|arXiv (Cornell University)|Dec 16, 2019

Limits and Structures in Graph Theory9 references8 citations

TL;DR

This paper establishes a new upper bound on the chromatic number of graphs with no odd $K_t$ minor, proving they are $O(t(\log t)^\beta)$-colorable for every $\beta > \frac{1}{4}$, improving upon the prior $O(t\sqrt{\log t})$ bound. The result extends recent advances in minor-free graph coloring using structural decomposition and linkage techniques in weakly connected graphs.

ABSTRACT

Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log t})$-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no $K_t$ minor, we make the first improvement on this bound by showing that every graph with no odd $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$.

Motivation & Objective

• To improve the upper bound on the chromatic number of graphs with no odd $K_t$ minor, which is central to the Odd Hadwiger's Conjecture.
• To extend recent progress on $K_t$-minor-free graphs to the odd minor setting, where the structure is more constrained.
• To establish a nearly-polynomial bound of the form $O(t(\log t)^\beta)$ for $\beta > 1/4$, closing the gap toward the conjectured $(t-1)$-colorability.
• To develop and apply a novel notion of weak connectivity and rooted linkages to manage structural complexity in graphs without odd $K_t$ minors.

Proposed method

• Introduce a new connectivity concept—weak $l$-connectivity—to control structural decomposition in graphs with no odd $K_t$ minor.
• Use a recursive coloring strategy based on finding subgraphs with high weak connectivity and large chromatic number.
• Apply Menger’s theorem and linkage theorems to construct vertex-disjoint paths and reroute them via $K_{2x}$-expansions in subgraphs.
• Construct a bipartite $K_{xy}$-expansion in the graph by combining rerouted linkages and trees rooted at specific vertices.
• Use parity-knitted sets and $8k$-linkedness in a central subgraph to embed a large complete bipartite minor.
• Combine all components into a single bipartite subgraph that supports a $K_{xy}$-expansion, implying high chromatic number.

Experimental results

Research questions

• RQ1 Can the chromatic number of graphs with no odd $K_t$ minor be bounded below the $O(t\sqrt{\log t})$ threshold established by Geelen et al.?
• RQ2 To what extent can techniques developed for $K_t$-minor-free graphs be adapted to the odd minor setting?
• RQ3 Is it possible to achieve a bound of the form $O(t(\log t)^\beta)$ for $\beta > 1/4$ in the odd minor case, matching the best known bound for $K_t$-minor-free graphs?
• RQ4 How can weak connectivity and rooted linkages be used to construct large bipartite minors in graphs without odd $K_t$ minors?

Key findings

• The chromatic number of any graph with no odd $K_t$ minor is bounded by $O(t(\log t)^\beta)$ for every $\beta > \frac{1}{4}$, improving the prior $O(t\sqrt{\log t})$ bound.
• The result confirms a significant step toward the Odd Hadwiger’s Conjecture, which posits $(t-1)$-colorability for such graphs.
• The proof introduces a new notion of weak connectivity that enables structural decomposition in graphs without odd $K_t$ minors.
• A key technical innovation is the construction of a bipartite $K_{xy}$-expansion using rerouted linkages and parity-knitted sets in a central subgraph.
• The method successfully extends recent advances in $K_t$-minor-free graph coloring to the more complex odd minor setting.
• The bound $O(t(\log t)^\beta)$ for $\beta > 1/4$ matches the best known bound for $K_t$-minor-free graphs, suggesting tightness in the current approach.