Regularization and asymmetric extremal numbers of subdivisions

TL;DR

This paper enhances the regularization theorem for graphs, establishing bounds on bipartite graphs avoiding certain subdivisions, with implications for extremal graph theory and bipartite subdivision problems.

Contribution

It develops an improved regularization theorem for graphs and bipartite graphs, providing tight bounds on edge counts avoiding specific subdivisions.

Findings

Enhanced regularization theorem with degree and average degree bounds

Upper bounds on edges avoiding bipartite subdivisions of complete graphs

Bounds are tight for infinitely many pairs (m,n)

Abstract

Given a real $\mu\geq 1$, a graph $H$ is $\mu$-almost-regular if $\Delta(H)\leq \mu \delta(H)$. The celebrated regularization theorem of Erd\H{o}s and Simonovits states that for every real $0<\varepsilon<1$ there exists a real $\mu=\mu(\varepsilon)$ such that every $n$-vertex graph $G$ with $\Omega(n^{1+\varepsilon})$ edges contains an $m$-vertex $\mu$-almost-regular subgraph $H$ with $\Omega(m^{1+\varepsilon})$ edges for some $n^{\varepsilon\frac{1-\varepsilon}{1+\varepsilon}}\leq m\leq n$. We develop an enhanced version of it in which the subgraph $H$ also has average degree at least $\Omega(\frac{d(G)}{\log n})$, where $d(G)$ is the average degree of $G$. We then give a bipartite analogue of the enhanced regularization theorem. Using the bipartite regularization theorem, we establish upper bounds on the maximum number of edges in a bipartite graph with part sizes $m$ and $n$ that does not contain a $2k$-subdivision of $K_{s,t}$ or $2k$-multi-subdivisions of $K_p$, thus extending the corresponding work of Janzer to the bipartite setting for even subdivisions. We show these upper bounds are tight up to a constant factor for infinitely many pairs $(m,n)$. The problem for estimating the maximum number of edges in a bipartite graph with part sizes $m$ and $n$ that does not contain a $(2k+1)$-subdivision of $K_{s,t}$ remains open.