Long Cycles in 2-Connected Tough Graphs
Abstract
Let G be a graph. The circumference of G, denoted by cir(G), is the length of a longest cycle in G, or zero if G is acyclic. In 1993, Broersma, van den Heuvel, Jung, and Veldman conjectured that, for every t>0, there is a constant A=A(t)>0 such that every 2-connected t-tough graph of order n has circumference at least Alog n; the conjecture is recorded as Conjecture~2 in the 2006 survey on toughness by Bauer, Broersma, and Schmeichel. In this note, we confirm the conjecture. More precisely, every 2-connected t-tough graph G of order n satisfies cir(G)≥ ⌈ logk((k−1)n+1)⌉, where k=⌈ 1/t⌉+2. The proof combines Win's bounded-degree spanning tree theorem with the theorem of Briański, Joret, Majewski, Micek, Seweryn, and Sharma that the treedepth of a 2-connected graph is at most its circumference.
Related
BibTeX
@misc{shan2026long,
title = {{Long Cycles in 2-Connected Tough Graphs}},
author = {Songling Shan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12135},
eprint = {2609.12135},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12135},
}
RIS
TY - UNPB
TI - Long Cycles in 2-Connected Tough Graphs
AU - Songling Shan
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12135
UR - https://arxiv.org/abs/2609.12135
AB - Let G be a graph. The circumference of G, denoted by cir(G), is the length of a longest cycle in G, or zero if G is acyclic. In 1993, Broersma, van den Heuvel, Jung, and Veldman conjectured that, for every t>0, there is a constant A=A(t)>0 such that every 2-connected t-tough graph of order n has circumference at least Alog n; the conjecture is recorded as Conjecture~2 in the 2006 survey on toughness by Bauer, Broersma, and Schmeichel. In this note, we confirm the conjecture. More precisely, every 2-connected t-tough graph G of order n satisfies cir(G)≥ ⌈ log_k((k−1)n+1)⌉, where k=⌈ 1/t⌉+2. The proof combines Win's bounded-degree spanning tree theorem with the theorem of Briański, Joret, Majewski, Micek, Seweryn, and Sharma that the treedepth of a 2-connected graph is at most its circumference.
ER -