{'fraction_non_alphanumeric': 0.04060913705583756, 'fraction_numerical': 0.024111675126903553, 'mean_word_length': 4.441379310344828, '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}}
The width of quadrangulations of the projective plane

We show that every 4‐chromatic graph on n vertices, with no two vertex‐disjoint odd cycles, has an odd cycle of length at most 12(1+8n−7) . Let G be a nonbipartite quadrangulation of the projective plane on n vertices. Our result immediately implies that G has edge‐width at most 12(1+8n−7) , which is sharp for infinitely many values of n. We also show that G has face‐width (equivalently, contains an odd cycle transversal of cardinality) at most 14(1+16n−15) , which is a constant away from the optimal; we prove a lower bound of n . Finally, we show that G has an odd cycle transversal of size at most 2Δn inducing a single edge, where Δ is the maximum degree. This last result partially answers a question of Nakamoto and Ozeki.