Combinatorics · Oct 6, 2026

Graph coloring, clique minors, and Colin de Verdière invariants

Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy

\chi_f(G)\gt h(G)

, where

h(G)

is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound

\chi_f(G)\le\mu(G)+1

. In the positive direction, every finite nonempty graph satisfies

\chi_{\mathrm{list}}(G)\le C h(G)

for a universal constant C.

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