Balanced clique subdivisions and cycles lengths in $K_{s,t}$-free graphs

Abstract

Let $t\ge s\ge 2$ be integers. Confirming a conjecture of Mader, Liu and Montgomery [J. Lond. Math. Soc., 2017] showed that every $K_{s,t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $\Omega (d^{\frac{s}{2(s-1)}})$ vertices. We give an improvement by showing that such a graph contains a balanced subdivision of a clique with the same order, where a balanced subdivision is a subdivision in which each edge is subdivided the same number of times. In 1975, Erdős asked whether the sum of the reciprocals of the cycle lengths in a graph with infinite average degree $d$ is necessarily infinite. Recently, Liu and Montgomery [J. Amer. Math. Soc., 2023] confirmed the asymptotically correct lower bound on the reciprocals of the cycle lengths, and provided a lower bound of at least $(\frac{1}{2}-o_d(1))\log d$. In this paper, we improve this low bound to $\left(\frac{s}{2(s-1)}-o_d(1)\right)\log d$ for $K_{s,t}$-free graphs. Both proofs of our results use the graph sublinear expansion property as well as some novel structural techniques.

arXiv admin note: text overlap with arXiv:2010.15802 by other authors