Full subgraphs

TL;DR

This paper investigates the size of large subgraphs with high minimum degree in dense graphs, providing new bounds, tightness results for specific densities, and exploring properties in random and pseudo-random graphs.

Contribution

It establishes a new lower bound on the size of full subgraphs for certain densities and demonstrates tightness of these bounds for specific cases, advancing understanding of dense graph structures.

Findings

New lower bound for full subgraph size in graphs with density p

Tightness of bounds shown for specific densities near rational numbers

Either a graph or its complement contains a large full subgraph of size Omega(n/log n)

Abstract

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erd\H{o}s, \L uczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a largest full subgraph of $G$. If $p\binom{n}{2}$ is a non-negative integer, define \[ f(n,p) = \min\{f(G) : \vert V(G)\vert = n, \ \vert E(G)\vert = p\binom{n}{2} \}.\] Erd\H{o}s, \L uczak and Spencer proved that for $n \geq 2$, \[ (2n)^{\frac{1}{2}} - 2 \leq f(n, {\frac{1}{2}}) \leq 4n^{\frac{2}{3}}(\log n)^{\frac{1}{3}}.\] In this paper, we prove the following lower bound: for $n^{-\frac{2}{3}} <p_n <1-n^{-\frac{1}{7}}$, \[ f(n,p) \geq \frac{1}{4}(1-p)^{\frac{2}{3}}n^{\frac{2}{3}} -1.\] Furthermore we show that this is tight up to a multiplicative constant factor for infinitely many $p$ near the elements of $\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\dots\}$. In contrast, we show that for any $n$-vertex graph $G$, either $G$ or $G^c$ contains a full subgraph on $\Omega(\frac{n}{\log n})$ vertices. Finally, we discuss full subgraphs of random and pseudo-random graphs, and several open problems.