Related papers: Degree conditions for matchability in $3$-partite …
The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…
We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $\Delta(F_i)\leq \Delta$ hold…
We find Dirac-type sufficient conditions for a hypergraph $\mathcal H$ with few edges to be hamiltonian. We also show that these conditions provide that $\mathcal H$ is {\em super-pancyclic}, i.e., for each $A \subseteq V(\mathcal H)$ with…
Erd\H{o}s posed the problem of finding conditions on a graph $G$ that imply the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We generalize this problem to general…
The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…
A hypergraph is said to be $1$-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of $1$-Sperner hypergraphs and their structure to graphs. In particular, we…
In 2001, Koml\'os, S\'ark\"ozy, and Szemer\'edi proved that every sufficiently large $n$-vertex graph with minimum degree at least $\left(1/2+\gamma\right)n$ contains all spanning trees with maximum degree at most $cn/\log n$. We extend…
Erd\H{o}s conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n/2 \rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least…
Given an $n\times n$ array $M$ ($n\ge 7$), where each cell is colored in one of two colors, we give a necessary and sufficient condition for the existence of a partition of $M$ into $n$ diagonals, each containing at least one cell of each…
Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by $n$ colours with at least $n + 1$ edges of each colour there must be a matching that uses each colour exactly once. In this paper we consider…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares. Since then rainbow structures have…
A celebrated conjecture of Sidorenko and Erd\H{o}s-Simonovits states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge…
A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices.…
The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{k,k}$ as a subgraph. A classical theorem due to K\H{o}v\'ari, S\'os, and Tur\'an…
In this note, we prove that there exists a universal constant $c=\frac{43}{50}$ such that for every $k\in \mathbb{N}$ and every $d<k/2$, every $k$-uniform hypergraph on $n$ vertices and with minimum $d$-degree at least…
We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For k greater than or equal to 2, a bipartite digraph D with colour classes of cardinalities k is hamiltonian if the sum of degrees of vertices u and…
In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + \gamma)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose…
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 $c$ be an edge-coloring of the complete $n$-vertex graph $K_n$. The problem of finding properly colored and rainbow Hamilton cycles in $c$ was initiated in 1976 by Bollob\'as and Erd\H os and has been extensively studied since then.…
A result of Balogh, Csaba, Jing and Pluh\'ar yields the minimum degree threshold that ensures a $2$-coloured graph contains a perfect matching of significant colour-bias (i.e., a perfect matching that contains significantly more than half…