Maximal Chordal Subgraphs

TL;DR

This paper determines the maximum size of a chordal subgraph guaranteed in any graph with given vertices and edges, solving a longstanding conjecture and providing asymptotic results.

Contribution

It proves a conjecture from the 1980s and precisely determines the function f(n,m) for all m up to n^2/3+1, advancing understanding of chordal subgraphs.

Findings

Proved the conjecture on f(n,m) for the case m = n^2/3+1.

Determined f(n,m) asymptotically for all m.

Exactly computed f(n,m) for m ≤ n^2/3+1.

Abstract

A chordal graph is a graph with no induced cycles of length at least $4$. Let $f(n,m)$ be the maximal integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. In 1985 Erd\H{o}s and Laskar posed the problem of estimating $f(n,m)$. In the late '80s, Erd\H{o}s, Gy\'arf\'as, Ordman and Zalcstein determined the value of $f(n,n^2/4+1)$ and made a conjecture on the value of $f(n,n^2/3+1)$. In this paper we prove this conjecture and answer the question of Erd\H{o}s and Laskar, determining $f(n,m)$ asymptotically for all $m$ and exactly for $m \leq n^2/3+1$.