Almost partitioning every 22-edge-coloured complete kk-graph into kk monochromatic tight cycles

A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$ vertex-disjoint monochromatic tight cycles that together cover $n - o(n)$ vertices

Towards Lehel's conjecture for 4-uniform tight cycles

A $k$-uniform tight cycle is a $k$-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size $k$ formed by $k$ consecutive vertices in the ordering. We prove that every red-blue edge-coloured $K_n^{(4)}$ contains a red and a blue tight cycle that are vertex-disjoint and together cover $n-o(n)$ vertices. Moreover, we prove that every red-blue edge-coloured $K_n^{(5)}$ contains four monochromatic tight cycles that are vertex-disjoint and together cover $n-o(n)$ vertices.Comment: arXiv admin note: text overlap with arXiv:1606.05616 by other author

Paths and cycles in graphs and hypergraphs

In this thesis we present new results in graph and hypergraph theory all of which feature paths or cycles. A $k$-uniform tight cycle $C^{(k)}_n$ is a $k$-uniform hypergraph on $n$ vertices with a cyclic ordering of its vertices such that the edges are all $k$-sets of consecutive vertices in the ordering. We consider a generalisation of Lehel's Conjecture, which states that every 2-edge-coloured complete graph can be partitioned into two cycles of distinct colour, to $k$-uniform hypergraphs and prove results in the 4- and 5-uniform case. For a $k$-uniform hypergraph~$H$, the Ramsey number ${r(H)}$ is the smallest integer $N$ such that any 2-edge-colouring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. We determine the Ramsey number for 4-uniform tight cycles asymptotically in the case where the length of the cycle is divisible by 4, by showing that $r(C^{(4)}_n)$ = (5+$o$(1))$n$. We prove a resilience result for tight Hamiltonicity in random hypergraphs. More precisely, we show that for any $\gamma$ >0 and $k$ $\geq$ 3 asymptotically almost surely, every subgraph of the binomial random $k$-uniform hypergraph $G^{(k)}(n, n^{\gamma -1})$ in which all $(k-1)$-sets are contained in at least $(\frac{1}{2}+2\gamma)pn$ edges has a tight Hamilton cycle. A random graph model on a host graph $H$ is said to be 1-independent if for every pair of vertex-disjoint subsets $A,B$ of $E(H)$, the state of edges (absent or present) in $A$ is independent of the state of edges in $B$. We show that $p$ = 4 - 2$\sqrt{3}$ is the critical probability such that every 1-independent graph model on $\mathbb{Z}^2 \times K_n$ where each edge is present with probability at least $p$ contains an infinite path

Perfect matchings in random sparsifications of Dirac hypergraphs

For all integers $n \geq k > d \geq 1$, let $m_{d}(k,n)$ be the minimum integer $D \geq 0$ such that every $k$-uniform $n$-vertex hypergraph $\mathcal H$ with minimum $d$-degree $\delta_{d}(\mathcal H)$ at least $D$ has an optimal matching. For every fixed integer $k \geq 3$, we show that for $n \in k \mathbb{N}$ and $p = \Omega(n^{-k+1} \log n)$, if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $\delta_{k-1}(\mathcal H) \geq m_{k-1}(k,n)$, then a.a.s.\ its $p$-random subhypergraph $\mathcal H_p$ contains a perfect matching ($m_{k-1}(k,n)$ was determined by R\"{o}dl, Ruci\'nski, and Szemer\'edi for all large $n \in k\mathbb N$). Moreover, for every fixed integer $d < k$ and $\gamma > 0$, we show that the same conclusion holds if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $\delta_d(\mathcal H) \geq m_{d}(k,n) + \gamma\binom{n - d}{k - d}$. Both of these results strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as "robust" versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, $\mathcal H$ has at least $\exp((1-1/k)n \log n - \Theta (n))$ many perfect matchings, which is best possible up to a $\exp(\Theta(n))$ factor.Comment: 25 pages + 2 page appendix; Theorem 1.5 was proved in independent work of Pham, Sah, Sawhney, and Simkin (arxiv:2210.03064

1-independent percolation on ℤ2×Kn

A random graph model on a host graph (Formula presented.) is said to be 1-independent if for every pair of vertex-disjoint subsets (Formula presented.) of (Formula presented.), the state of edges (absent or present) in (Formula presented.) is independent of the state of edges in (Formula presented.). For an infinite connected graph (Formula presented.), the 1-independent critical percolation probability (Formula presented.) is the infimum of the (Formula presented.) such that every 1-independent random graph model on (Formula presented.) in which each edge is present with probability at least (Formula presented.) almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that (Formula presented.) tends to a limit in (Formula presented.) as (Formula presented.), and they asked for the value of this limit. We make progress on a related problem by showing that (Formula presented.) In fact, we show that the equality above remains true if the sequence of complete graphs (Formula presented.) is replaced by a sequence of weakly pseudorandom graphs on (Formula presented.) vertices with average degree (Formula presented.). We conjecture the answer to Balister and Bollobás's question is also (Formula presented.)