Hadwiger's Conjecture

Theorem 1.1: there are arbitrarily large $m$ and $m$-vertex graphs with $\alpha(G)\le2$ and connected-matching number below $m/100$. With the counting bound $h(G)\le(|V(G)|+4\,\mathrm{cm}(G)+2)/3$, Corollary 1.2 gives $h(G)<26m/75+2/3<m/2\le\chi_f(G)\le\chi(G)$. So Hadwiger's conjecture fails, already in the case $\alpha\le2$ and even for fractional colouring, with $\chi/h$ above about $1.44$ in the limit. The graphs are complements of a triangle-free hole relation built from tensor frames over $\mathbb F_2$, so $\alpha\le2$; the four-hole conflicts that kill large connected matchings are forced by a distribution theorem and a container argument. It does not say anything about small $t$, and it does not refute linear bounds $\chi\le C\,h$, which the companion list-colouring paper proves.

Theorem 1.1: there are arbitrarily large $m$ and $m$-vertex graphs with $\alpha(G)\le2$ and connected-matching number below $m/100$. With the counting bound $h(G)\le(|V(G)|+4\,\mathrm{cm}(G)+2)/3$, Corollary 1.2 gives $h(G)<26m/75+2/3<m/2\le\chi_f(G)\le\chi(G)$. So Hadwiger's conjecture fails, already in the case $\alpha\le2$ and even for fractional colouring, with $\chi/h$ above about $1.44$ in the limit. The graphs are complements of a triangle-free hole relation built from tensor frames over $\mathbb F_2$, so $\alpha\le2$; the four-hole conflicts that kill large connected matchings are forced by a distribution theorem and a container argument. It does not say anything about small $t$, and it does not refute linear bounds $\chi\le C\,h$, which the companion list-colouring paper proves.