A linear-time algorithm for finding a complete graph minor in a dense graph

TL;DR

This paper presents a linear-time algorithm to find a complete graph minor in dense graphs, improving previous algorithms by reducing the degree threshold required for the minor’s existence.

Contribution

It introduces a linear-time algorithm for finding K_t minors in dense graphs with degree thresholds close to the theoretical minimum.

Findings

The algorithm works in linear time for graphs with degree at least (2+ε)g(t).

It improves previous algorithms that required degree at least 2^{t-2}.

The result applies to sufficiently large t and fixed ε > 0.

Abstract

Let g(t) be the minimum number such that every graph G with average degree d(G) \geq g(t) contains a K_{t}-minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that g(t) \in \Theta(t*sqrt{log t}). This article shows that for all fixed \epsilon > 0 and fixed sufficiently large t \geq t(\epsilon), if d(G) \geq (2+\epsilon)g(t) then we can find this K_{t}-minor in linear time. This improves a previous result by Reed and Wood who gave a linear-time algorithm when d(G) \geq 2^{t-2}.