Longest Cycle above Erd\H{o}s-Gallai Bound
Data Structures and Algorithms
2022-02-08 v1 Discrete Mathematics
Combinatorics
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.
Cite
@article{arxiv.2202.03061,
title = {Longest Cycle above Erd\H{o}s-Gallai Bound},
author = {Fedor V. Fomin and Petr A. Golovach and Danil Sagunov and Kirill Simonov},
journal= {arXiv preprint arXiv:2202.03061},
year = {2022}
}