Related papers: Orientations making k-cycles cyclic

We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $\delta(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $\delta(G)$ edges…

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 $k$…

We say that a k-uniform hypergraph C is an l-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of…

Let $k$ be a positive integer. A $k$-cycle-factor of an oriented graph is a set of disjoint cycles of length $k$ that covers all vertices of the graph. In this paper, we prove that there exists a positive constant $c$ such that for $n$…

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 $k$…

Building on previous work by Cameron et al. in [3], we give a recurrence for computing the number of acyclic orientations of complete $k$-partite graphs, which can be implemented to obtain a dynamic programming algorithm running in time…

An oriented graph is a directed graph which can be obtained from a simple undirected graph by orienting its edges. In this paper we show that any oriented graph G on n vertices with minimum indegree and outdegree at least (1/2-o(1))n…

We investigate the minimum number of cycles of specified lengths in planar $n$-vertex triangulations $G$. It is proven that this number is $\Omega(n)$ for any cycle length at most $3 + \max \{ {\rm rad}(G^*), \lceil…

A $k$-cycle in a graph is a cycle of length $k.$ A graph $G$ of order $n$ is called edge-pancyclic if for every integer $k$ with $3\le k\le n,$ every edge of $G$ lies in a $k$-cycle. It seems difficult to determine the minimum size $f(n)$…

An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set.…

An edge of a graph of order $n$ is pancyclic if it lies in a cycle of every length $3,\ldots,n$. A graph of order $n$ is vertex-pancyclic if every vertex lies in a cycle of every length $3,\ldots,n$. Recently, Li and Zhan proved that every…

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…

Answering a question by Letzter and Snyder, we prove that for large enough $k$ any $n$-vertex graph $G$ with minimum degree at least $\frac{1}{2k-1}n$ and without odd cycles of length less than $2k+1$ is $3$-colourable. In fact, we prove a…

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…

For $0\leq \ell <k$, a Hamiltonian $\ell$-cycle in a $k$-uniform hypergraph $H$ is a cyclic ordering of the vertices of $H$ in which the edges are segments of length $k$ and every two consecutive edges overlap in exactly $\ell$ vertices. We…

A pancyclic graph is a simple graph containing a cycle of length $k$ for all $3\leq k\leq n$. Let $m(n)$ be the minimum number of edges of all pancyclic graphs on $n$ vertices. Exact values are given for $m(n)$ for $n\leq 37$, combining…

A graph $G$ of order $n$ is called edge-pancyclic if, for every integer $k$ with $3 \leq k \leq n$, every edge of $G$ lies in a cycle of length $k$. Determining the minimum size $f(n)$ of a simple edge-pancyclic graph with $n$ vertices…

A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least log n/(100log k), improving a bound of Alon,…

The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions:…

A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. R\"{o}dl, Ruci\'{n}ski, and Szemer\'{e}di proved…