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

Abstract

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor 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. We show that every graph with no $K_t$ minor is $O(t(\log t{)}^{\beta })$-colorable for every $\beta >1/4$, making the first improvement on the order of magnitude of the Kostochka-Thomason bound.

arXiv commentThis version adds a new coauthor and significantly strengthens the main result by combining the previous version with arXiv:1911.01491. Several other major changes in content and presentation are made

References in corpus (4)

• Halfway to Hadwiger's Conjecture
• A new upper bound on the chromatic number of graphs with no odd $K_t$ minor
• Connectivity and choosability of graphs with no $K_t$ minor
• A Note on Hadwiger's Conjecture

Cited by in corpus (12)

• Extremal density for sparse minors and subdivisions
• An even better Density Increment Theorem and its application to Hadwiger's Conjecture
• Clustered Coloring of Graphs Excluding a Subgraph and a Minor
• Further progress towards Hadwiger's conjecture
• Clique immersions and independence number
• Halfway to Hadwiger's Conjecture
• A new upper bound on the chromatic number of graphs with no odd $K_t$ minor
• Further Progress towards the List and Odd Versions of Hadwiger's Conjecture
• Immersion and clustered coloring
• Clique minors in graphs with a forbidden subgraph
• Clique immersion in graphs without fixed bipartite graph
• Improved bound for Hadwiger's conjecture