Author

Freddie Illingworth

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Preprint Oct 2026

Hypergraphs with the smallest Hall ratio are still hard to colour

For $H$ a hypergraph, the Hall ratio $\rho(H)=\text{max}_{X\subseteq V(H)}\frac{\lvert{X}\rvert}{\alpha(H[X])}$ is a natural lower bound for its chromatic number, and among non-empty $k$-uniform hypergraphs the smallest possible Hall ratio is $k/(k-1)$. For graphs ($k = 2$), having minimum Hall ratio $2$ forces chromat...

V. Bitonti, Freddie Illingworth, Pedram Noohi et al. · 0 citations

Preprint Sep 2026

Disproof of the dominating Hadwiger conjecture

Hadwiger's conjecture (1943) states that every graph $G$ with chromatic number at least $t$ contains a $K_t$-model: a collection of $t$ vertex-disjoint connected subgraphs $T_1,\dots,T_t$ such that for all $1\le i<j\le t$ some vertex in $T_j$ has a neighbour in $T_i$. Replacing"some"in this definition by"every"gives ri...

Freddie Illingworth, Raphael Steiner · 1 citation