Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 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

Powers of Hamilton cycles in pseudorandom graphs (1402.0984v2)

Published 5 Feb 2014 in math.CO

Abstract: We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph $G$ is $(\varepsilon,p,k,\ell)$-pseudorandom if for all disjoint $X$ and $Y\subset V(G)$ with $|X|\ge\varepsilon pkn$ and $|Y|\ge\varepsilon p\ell n$ we have $e(X,Y)=(1\pm\varepsilon)p|X||Y|$. We prove that for all $\beta>0$ there is an $\varepsilon>0$ such that an $(\varepsilon,p,1,2)$-pseudorandom graph on $n$ vertices with minimum degree at least $\beta pn$ contains the square of a Hamilton cycle. In particular, this implies that $(n,d,\lambda)$-graphs with $\lambda\ll d{5/2 }n{-3/2}$ contain the square of a Hamilton cycle, and thus a triangle factor if $n$ is a multiple of $3$. This improves on a result of Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403--426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.

Citations (20)

Summary

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