Rainbow clique subdivisions

We show that for any integer $t \ge 2$, every properly edge colored $n$-vertex graph with average degree at least $(\log n)^{2+o(1)}$ contains a rainbow subdivision of a complete graph of size $t$. Note that this bound is within a log factor of the lower bound. This also implies a result on the rainbow Tur\'{a}n number of cycles

Disjoint isomorphic balanced clique subdivisions

A classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least has a subdivision of , the complete graph on k vertices. We study two directions extending this result. • Verstraëte conjectured that a quadratic bound guarantees in fact two vertex-disjoint isomorphic copies of a -subdivision. • Thomassen conjectured that for each there is some such that every graph with average degree at least d contains a balanced subdivision of . Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on remains open. In this paper, we show that a quadratic lower bound on average degree suffices to force a balanced -subdivision. This gives the right order of magnitude of the optimal needed in Thomassen's conjecture. Since a balanced -subdivision trivially contains m vertex-disjoint isomorphic -subdivisions, this also confirms Verstraëte's conjecture in a strong sense

Sublinear expanders and their applications

In this survey we aim to give a comprehensive overview of results using sublinear expanders. The term sublinear expanders refers to a variety of definitions of expanders, which typically are defined to be graphs $G$ such that every not-too-small and not-too-large set of vertices $U$ has neighbourhood of size at least $\alpha |U|$, where $\alpha$ is a function of $n$ and $|U|$. This is in contrast with linear expanders, where $\alpha$ is typically a constant. :We will briefly describe proof ideas of some of the results mentioned here, as well as related open problems.Comment: 39 pages, 15 figures. This survey will appear in `Surveys in Combinatorics 2024' (the proceedings of the 30th British Combinatorial Conference

A solution to Erd\H{o}s and Hajnal's odd cycle problem

In 1981, Erd\H{o}s 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 $\mathcal{C}(G)$ be the set of cycle lengths in a graph $G$ and let $\mathcal{C}_\text{odd}(G)$ be the set of odd numbers in $\mathcal{C}(G)$. We prove that, if $G$ has chromatic number $k$, then $\sum_{\ell\in \mathcal{C}_\text{odd}(G)}1/\ell\geq (1/2-o_k(1))\log k$. This solves Erd\H{o}s and Hajnal's odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erd\H{o}s asked whether there is some $d$ such that each graph with chromatic number at least $d$ (or perhaps even only average degree at least $d$) 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 $k$, there is some $d$ so that every graph with average degree at least $d$ has a subdivision of the complete graph $K_k$ in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984.Comment: 42 pages, 3 figures. Version accepted for publicatio

A solution to Erdős and Hajnal’s odd cycle problem

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) be the set of cycle lengths in a graph G and let Codd(G) be the set of odd numbers in C(G). We prove that, if G has chromatic number k, then ∑ℓ∈Codd(G)1/ℓ≥(1/2−ok(1))logk. 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 such that each graph with chromatic number at least d (or perhaps even only average degree at least 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, there is some d so that every graph with average degree at least d has a subdivision of the complete graph Kk in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984