A solution to Erdős and Hajnal’s odd cycle problem
Journal of the American Mathematical Society(2023)
摘要
In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let C ( G ) \mathcal {C}(G) be the set of cycle lengths in a graph G G and let C o d d ( G ) \mathcal {C}_{\mathrm {odd}}(G) be the set of odd numbers in C ( G ) \mathcal {C}(G) . We prove that, if G G has chromatic number k k , then ∑ ℓ ∈ C o d d ( G ) 1 / ℓ ≥ ( 1 / 2 − o k ( 1 ) ) log k \sum _{\ell \in \mathcal {C}_{\mathrm {odd}}(G)}1/\ell \geq (1/2-o_k(1))\log k . This solves Erdős and Hajnal’s odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erdős asked whether there is some d d such that each graph with chromatic number at least d d (or perhaps even only average degree at least d d ) has a cycle whose length is a power of 2. We show that an average degree condition is sufficient for this problem, solving it with methods that apply to a wide range of sequences in addition to the powers of 2. Finally, we use our methods to show that, for every k k , there is some d d so that every graph with average degree at least d d has a subdivision of the complete graph K k K_k in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984.
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关键词
cycle,erdős,hajnals
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