{'fraction_non_alphanumeric': 0.04706927175843695, 'fraction_numerical': 0.028419182948490232, 'mean_word_length': 4.635, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Toroidal graphs containing neither K5− nor 6‐cycles are 4‐choosable

The choosability χℓ(G) of a graph G is the minimum k such that having k colors available at each vertex guarantees a proper coloring. Given a toroidal graph G, it is known that χℓ(G)≤7 , and χℓ(G)=7 if and only if G contains K7. Cai et al. (J Graph Theory 65(1) (2010), 1–15) proved that a toroidal graph G without 7‐cycles is 6‐choosable, and χℓ(G)=6 if and only if G contains K6. They also proved that a toroidal graph G without 6‐cycles is 5‐choosable, and conjectured that χℓ(G)=5 if and only if G contains K5. We disprove this conjecture by constructing an infinite family of non‐4‐colorable toroidal graphs with neither K5 nor cycles of length at least 6; moreover, this family of graphs is embeddable on every surface except the plane and the projective plane. Instead, we prove the following slightly weaker statement suggested by Zhu: toroidal graphs containing neither K5− (a K5 missing one edge) nor 6‐cycles are 4‐choosable. This is sharp in the sense that forbidding only one of the two structures does not ensure that the graph is 4‐choosable.