Related papers: Unavoidable patterns in locally balanced colouring…

A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of…

A graph/multigraph $G$ is locally irregular if endvertices of every its edge possess different degrees. The locally irregular edge coloring of $G$ is its edge coloring with the property that every color induces a locally irregular…

The rainbow Tur\'an number $\mathrm{ex}^*(n,H)$ of a graph $H$ is the maximum possible number of edges in a properly edge-coloured $n$-vertex graph with no rainbow subgraph isomorphic to $H$. We prove that for any integer $k\geq 2$,…

It is shown that a hamiltonian $n/2$-regular bipartite graph $G$ of order $2n>8$ contains a cycle of length $2n-2$. Moreover, if such a cycle can be chosen to omit a pair of adjacent vertices, then $G$ is bipancyclic.

We show that every graph $G$ on $n$ vertices with $\delta(G) \geq (1/2+\varepsilon)n$ is spanned by a complete blow-up of a cycle with clusters of nearly uniform size $\Omega(\log n)$. The proof is based on a recently introduced approach…

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption…

We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of $K_{n,n}$ with $2n/3 + o(n)$ colors such that each $4$-cycle…

We show that for all $\ell, k, n$ with $\ell \leq k/2$ and $(k-\ell)$ dividing $n$ the following hypergraph-variant of Lehel's conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph $\mathcal{K}_n^{(k)}$ on $n$…

Let $K_n^c$ be an edge-coloured complete graph on $n$ vertices. Let $\Delta_{\rm mon}(K_n^c)$ denote the largest number of edges of the same colour incident with a vertex of $K_n^c$. A properly coloured cycle is a cycle such that no two…

A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of…

This paper examines vertex colorings of graphs with constraints on the distribution of colors in vertex neighborhoods. We introduce color 2-switches and color degree matrices. The color degree matrix of a $k$-colored graph is an analog of…

We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k \geq 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $\Omega(\sqrt{n})$. This improves on the…

For which values of $k$ does a uniformly chosen $3$-regular graph $G$ on $n$ vertices typically contain $ n/k$ vertex-disjoint $k$-cycles (a $k$-cycle factor)? To date, this has been answered for $k=n$ and for $k \ll \log n$; the former,…

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges is in each color class. If there exists an integer $k$ such that, for $n$ sufficiently…

For a graph $G$ with a list assignment $L$ and two $L$-colorings $\alpha$ and $\beta$, an $L$-recoloring sequence from $\alpha$ to $\beta$ is a sequence of proper $L$-colorings where consecutive colorings differ at exactly one vertex. We…

A $k$-coloring of a graph is an assignment of integers between $1$ and $k$ to vertices in the graph such that the endpoints of each edge receive different numbers. We study a local variation of the coloring problem, which imposes further…

It is proved that every connected graph $G$ on $n$ vertices with $\chi(G) \geq 4$ has at most $k(k-1)^{n-3}(k-2)(k-3)$ $k$-colourings for every $k \geq 4$. Equality holds for some (and then for every) $k$ if and only if the graph is formed…

Let $S=R\cup B$ be a point set in the plane in general position such that each of its elements is colored either red or blue, where $R$ and $B$ denote the points colored red and the points colored blue, respectively. A quadrilateral with…

In this paper, we investigate two questions on Kneser graphs $KG_{n,k}$. First, we prove that the union of $s$ intersecting families in ${[n]\choose k}$ has size at most ${n\choose k}-{n-s\choose k}$ for all sufficiently large $n$ that…

It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $\Delta^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional…