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Register a new userRegularity inheritance in hypergraphs
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We give a new approach to handling hypergraph regularity. This approach allows for vertex-by-vertex embedding into regular partitions of hypergraphs, and generalises to regular partitions of sparse hypergraphs. We also prove a corresponding sparse hypergraph regularity lemma.
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We show that a quasirandom $k$-uniform hypergraph $G$ has a tight Euler tour subject to the necessary condition that $k$ divides all vertex degrees. The case when $G$ is complete confirms a conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the $k$-subsets of an $n$-set.
Given integers $k,j$ with $1le j le k-1$, we consider the length of the longest $j$-tight path in the binomial random $k$-uniform hypergraph $H^k(n,p)$. We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the `Pathfinder algorithm, a depth-first search algorithm which discovers $j$-tight paths in a $k$-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm will find a long $j$-tight path.
We show that, for a natural notion of quasirandomness in $k$-uniform hypergraphs, any quasirandom $k$-uniform hypergraph on $n$ vertices with constant edge density and minimum vertex degree $Omega(n^{k-1})$ contains a loose Hamilton cycle. We also give a construction to show that a $k$-uniform hypergraph satisfying these conditions need not contain a Hamilton $ell$-cycle if $k-ell$ divides $k$. The remaining values of $ell$ form an interesting open question.
A hypergraph is a generalization of a graph where edges can connect any number of vertices. In this paper, we extend the study of locating-dominating sets to hypergraphs. Along with some basic results, sharp bounds for the location-domination number of hypergraphs in general and exact values with specified conditions are investigated. Moreover, locating-dominating sets in some specific hypergraphs are found.
We prove that for any integer $kgeq 2$ and $varepsilon>0$, there is an integer $ell_0geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+varepsilon)n$ has a fractional decomposition into tight cycles of length $ell$ ($ell$-cycles for short) whenever $ellgeq ell_0$ and $n$ is large in terms of $ell$. This is essentially tight. This immediately yields also approximate integral decompositions for these hypergraphs into $ell$-cycles. Moreover, for graphs this even guarantees integral decompositions into $ell$-cycles and solves a problem posed by Glock, Kuhn and Osthus. For our proof, we introduce a new method for finding a set of $ell$-cycles such that every edge is contained in roughly the same number of $ell$-cycles from this set by exploiting that certain Markov chains are rapidly mixing.
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