math.CO — Combinatorics

Every odd antihole is strongly chromatic-choosable

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove the complete Fiujlaali--McDonald conjecture on odd antiholes unconditionally: for every $\ell\ge3$ the only assignments of $(\ell-1)$-element lists from which the odd antihole $\cbar{2\ell-1}$ cannot be coloured are the constant ones. Consequently, an odd-hole-free $k$-critical graph of maximum degree $\Delta$ containing an odd antihole of length $2(\Delta-k)+7$ is that antihole.