Related papers: Four Blocks Cycles C(k,1,1,1) in Digraphs

Cohen et al. conjectured that for every oriented cycle $C$ there exist an integer $f(C)$ such that every strong $f(C)$-chromatic digraph contains a subdivision of $C$. El Joubbeh confirmed this conjecture for Hamiltonian digraphs. Indeed,…

A graph construction that produces a k-regular graph on n vertices for any choice of k >= 3 and n = m(k+1) for integer m >= 2 is described. The number of Hamiltonian cycles in such graphs can be explicitly determined as a function of n and…

We characterize all orientations of cycles $C$ for which for every fixed $\varepsilon > 0$ there exists a constant $c \geq 1$ such that every digraph $D$ without loops or parallel arcs with $\chi(D) \geq c$ and minimum out-degree at least…

A $3$-connected graph $G$ is essentially $4$-connected if, for any $3$-cut $S\subseteq V(G)$ of $G$, at most one component of $G-S$ contains at least two vertices. We prove that every essentially $4$-connected maximal planar graph $G$ on…

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…

Given a subgroup $H$ of a group $(G,+)$, a $(G,H,k,1)$ difference family (DF) is a set $\mathcal F$ of $k$-subsets of $G$ such that $\{f-f':f,f'\in F, f\neq f',F\in \mathcal F\}=G\setminus H$. Let $g\mathbb Z_{gh}$ is the subgroup of order…

Let $k$ and $\ell$ be positive integers. A cycle with two blocks $C(k,\ell)$ is a digraph consisting of two internally vertex disjoint directed paths of lengths $k$ and $\ell$ with the same initial vertex and terminal vertex.…

A 4-wheel is a graph formed by a cycle C and a vertex not in C that has at least four neighbors in C. We prove that a graph G that does not contain a 4-wheel as a subgraph is 4-colorable and we describe some structural properties of such a…

Let $L$ be a set of positive integers. We call a (directed) graph $G$ an $L$\emph{-cycle graph} if all cycle lengths in $G$ belong to $L$. Let $c(L,n)$ be the maximum number of cycles possible in an $n$-vertex $L$-cycle graph (we use…

For $k \ge 4$, a loose $k$-cycle $C_k$ is a hypergraph with distinct edges $e_1, e_2, \ldots, e_k$ such that consecutive edges (modulo $k$) intersect in exactly one vertex and all other pairs of edges are disjoint. Our main result is that…

The dicycle transversal number t(D) of a digraph D is the minimum size of a dicycle transversal of D, i. e. a set T of vertices of D such that D-T is acyclic. We study the following problem: Given a digraph D, decide if there is a dicycle B…

We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vec{\chi}}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph…

A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for every positive integer k, every triangle-free graph with sufficiently large chromatic number contains holes of k consecutive lengths.

A hole is an induced cycle of length at least 4. A graph is called a pentagraph if it has no cycles of length 3 or 4 and has no holes of odd length at least 7, and is called a heptagraph if it has no cycles of length less than 7 and has no…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

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 $k$…

Every 4-connected graph $G$ with minimum degree $\delta$ and connectivity $\kappa$ either contains a cycle of length at least $4\delta-\kappa-4$ or every longest cycle in $G$ is a dominating cycle.

We generalise a result of Corr\'{a}di and Hajnal and show that every graph with average degree at least $\tfrac{4}{3}kr$ contains $k$ vertex disjoint cycles, each of order at least $r$, as long as $k \geq 6$. This bound is sharp when $r=3$.

Let D be an acyclic orientation of a simple graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define m and M to be the minimum and the maximum number of…

In this note we prove: {\it Let $D$ be a 2-strong digraph of order $n$ such that its $n-1$ vertices have degrees at least $n+k$ and the remaining vertex $z$ has degree at least $n-k-4$, where $k$ is a positive integer. If $D$ contains a…