Related papers: Almost partitioning 2-coloured complete 3-uniform …
An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomass\'e, we prove that the vertex set of any $2$-locally…
In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$…
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C \subset H is a collection of N edges for which there is an ordering of the vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i, v_{i+1},…
We prove that any k-uniform hypergraph on n vertices with minimum degree at least n/(2(k-1))+o(n) contains a loose Hamilton cycle. The proof strategy is similar to that used by K\"uhn and Osthus for the 3-uniform case. Though some…
A question posed independently by Letzter and Pokrovskiy asks: how many vertex-disjoint monochromatic cycles are needed to cover the vertex set of an $r$-edge-coloured graph, as a function of its minimum (uncoloured) degree? We resolve this…
We show that every 3-uniform hypergraph with minimum vertex degree at least $0.8\binom{n-1}{2}$ contains a tight Hamiltonian cycle.
Gy\'arf\'as, Gy\H{o}ri and Simonovits proved that if a $3$-uniform hypergraph with $n$ vertices has no linear cycles, then its independence number $\alpha \ge \frac{2n} {5}$. The hypergraph consisting of vertex disjoint copies of a complete…
It has been conjectured that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every (r - 1)-coloring of the edges of the complete r-uniform hypergraph on n vertices. In this paper, we show that…
Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of 3-uniform loose paths when one of the paths is significantly larger than the other:…
It is a longstanding conjecture that every simple drawing of a complete graph on $n \geq 3$ vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to "there exists a crossing-free Hamiltonian path between each…
Erd\H{o}s, Gy\'arf\'as and Pyber showed that every $r$-edge-coloured complete graph $K_n$ can be covered by $25 r^2 \log r$ vertex-disjoint monochromatic cycles (independent of $n$). Here, we extend their result to the setting of binomial…
An edge-colored graph $G$ is called properly colored if every two adjacent edges are assigned different colors. A monochromatic triangle is a cycle of length 3 with all the edges having the same color. Given a tree $T_0$, let…
We show that if we color the hyperedges of the complete $3$-uniform complete graph on $2n+\sqrt{18n+1}+2$ vertices with $n$ colors, then one of the color classes contains a loose path of length three.
We prove that for any integer $k\geq 2$ and $\varepsilon>0$, there is an integer $\ell_0\geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+\varepsilon)n$ has a fractional decomposition into…
This work studies the hardness of finding independent sets in hypergraphs which are either 2-colorable or are almost 2-colorable, i.e. can be 2-colored after removing a small fraction of vertices and the incident hyperedges. To be precise,…
An $r$-uniform tight cycle of length $\ell>r$ is a hypergraph with vertices $v_1,\dots,v_\ell$ and edges $\{v_i,v_{i+1},\dots,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $\ell$. It was shown by Sudakov and Tomon that for each…
An edge-coloring of a hypergraph is {\em spanning} if every vertex sees every color used in the coloring. In this paper, we prove that for $k \geq 2r \geq 6$, in any spanning $k$-coloring of the edges of a complete $r$-partite $r$-uniform…
Given an $r$-edge-colouring of the edges of a graph $G$, we say that it can be partitioned into $p$ monochromatic cycles when there exists a set of $p$ vertex-disjoint monochromatic cycles covering all the vertices of $G$. In the literature…
We show that there exists an absolute constant $A$ such that for each $k\ge2$ and every coloring of the edges of the complete $k$-uniform hypergraph on $ Ar$ vertices with $r$ colors, one of the color classes contains a loose path of length…
Given a $k$-colouring of the edges of the complete graph $K_n$, are there $k-1$ monochromatic components that cover its vertices? This important special case of the well-known Lov\'asz-Ryser conjecture is still open. In this paper we…