Related papers: Minimum degree conditions for monochromatic cycle …

We show that every sufficiently large oriented graph with minimum in- and outdegree at least (3n-4)/8 contains a Hamilton cycle. This is best possible and solves a problem of Thomassen from 1979.

A well known generalisation of Dirac's theorem states that if a graph $G$ on $n\ge 4k$ vertices has minimum degree at least $n/2$ then $G$ contains a $2$-factor consisting of exactly $k$ cycles. This is easily seen to be tight in terms of…

The chromatic threshold delta_chi(H) of a graph H is the infimum of d>0 such that there exists C=C(H,d) for which every H-free graph G with minimum degree at least d|G| satisfies chi(G)<C. We prove that delta_chi(H) \in {(r-3)/(r-2),…

Let $G$ be a $2$-coloring of a complete graph on $n$ vertices, for sufficiently large $n$. We prove that $G$ contains at least $n^{(\frac{1}{4} - o(1))\log n}$ monochromatic complete subgraphs of size $r$, where \[ 0.3\log n < r < 0.7\log…

Let $G$ be a $t$-tough graph of order $n$ and minimum degree $\delta$ with $t>1$. It is proved that if $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.

For integers $k, r > 0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to at least $\min\{r, d(v)\}$ differently colored…

It is known that in any $r$-coloring of the edges of a complete $r$-uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on $n$ vertices, what is the largest monochromatic component one can…

We prove: (i) if $G$ is a 1-tough graph of order $n$ and minimum degree $\delta$ with $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle unless $G$ belongs to an easily specified class of graphs with $\kappa(G)=2$ and…

We show that the minimum number of orientations of the edges of the n-vertex complete graph having the property that every triangle is made cyclic in at least one of them is $\lceil\log_2(n-1)\rceil$. More generally, we also determine the…

In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We…

An edge-colored graph is a graph in which each edge is assigned a color. Such a graph is called strongly edge-colored if each color class forms an induced matching, and called rainbow if all edges receive pairwise distinct colors. In this…

Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this…

We investigate minimum vertex degree conditions for $3$-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections…

We prove that a random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, is typically such that for any $r$-colouring of its edges there exists a Hamilton cycle with at least $(2/(r+ 1)-o(1))n$ edges of the same colour. This…

For an integer $r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $min\{r, d(v)\}$ different colors.…

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…

We prove that every triangle-free graph with maximum degree $\Delta$ has list chromatic number at most $(1+o(1))\frac{\Delta}{\ln \Delta}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that…

We prove asymptotically optimal bounds on the number of edges a graph $G$ must have in order that any $r$-colouring of $E(G)$ has a colour class which contains every $D$-degenerate graph on $n$ vertices with bounded maximum degree. We also…

The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum…

We show that for any integer $t \ge 2$, every properly edge colored $n$-vertex graph with average degree at least $(\log n)^{2+o(1)}$ contains a rainbow subdivision of a complete graph of size $t$. Note that this bound is within $(\log…