Halfway to Hadwiger's Conjecture

Abstract

In 1943, Hadwiger conjectured that every $K_t$-minor-free graph is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Very recently, Norin and Song proved that every graph with no $K_t$ minor is $O(t(\log t{)}^{0.354})$-colorable. Improving on the second part of their argument, we prove that every graph with no $K_t$ minor is $O(t(\log t{)}^{\beta })$-colorable for every $\beta >\frac{1}{4}$.

Merged into arXiv:1910.09378v2

References in corpus (1)

• Breaking the degeneracy barrier for coloring graphs with no $K_t$ minor

Cited by in corpus (6)

• Breaking the degeneracy barrier for coloring graphs with no $K_t$ minor
• An even better Density Increment Theorem and its application to Hadwiger's Conjecture
• Further progress towards Hadwiger's conjecture
• Further Progress towards the List and Odd Versions of Hadwiger's Conjecture
• Immersion and clustered coloring
• Clique minors in graphs with a forbidden subgraph