Longest Cycle above Erd\H{o}s-Gallai Bound

TL;DR

This paper presents an algorithm that determines whether a 2-connected graph contains a cycle longer than the Erdős-Gallai bound by a specified amount, resolving an open problem in graph theory.

Contribution

It introduces a fixed-parameter tractable algorithm for finding long cycles relative to the Erdős-Gallai bound in 2-connected graphs, along with new graph-theoretical results.

Findings

Algorithm runs in 2^{O(k)} n^{O(1)} time for parameter k

Decides existence of cycles longer than ad(G)+k

Provides new insights into cycle length bounds

Abstract

In 1959, Erd\H{o}s and Gallai proved that every graph G with average vertex degree ad(G)\geq 2 contains a cycle of length at least ad(G). We provide an algorithm that for k\geq 0 in time 2^{O(k)} n^{O(1)} decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.