Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A hypergraph analog of Dirac's Theorem for long cycles in 2-connected graphs, II: Large uniformities (2310.13190v1)

Published 19 Oct 2023 in math.CO

Abstract: Dirac proved that each $n$-vertex $2$-connected graph with minimum degree $k$ contains a cycle of length at least $\min{2k, n}$. We obtain analogous results for Berge cycles in hypergraphs. Recently, the authors proved an exact lower bound on the minimum degree ensuring a Berge cycle of length at least $\min{2k, n}$ in $n$-vertex $r$-uniform $2$-connected hypergraphs when $k \geq r+2$. In this paper we address the case $k \leq r+1$ in which the bounds have a different behavior. We prove that each $n$-vertex $r$-uniform $2$-connected hypergraph $H$ with minimum degree $k$ contains a Berge cycle of length at least $\min{2k,n,|E(H)|}$. If $|E(H)|\geq n$, this bound coincides with the bound of the Dirac's Theorem for 2-connected graphs.

Summary

We haven't generated a summary for this paper yet.