Related papers: Linkedness and ordered cycles in digraphs

Dirac proved that each $n$-vertex $2$-connected graph with minimum degree $k$ contains a cycle of length at least $\min\{2k, n\}$. We obtain analogous results for Berge cycles in hypergraphs. Recently, the authors proved an exact lower…

Given two $k$-dicolourings of a digraph $D$, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for $k=2$ and for…

A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k distinct vertices v_1, ..., v_k of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing these k vertices in the specified…

The girth of a graph $G$ is the length of a shortest cycle of $G$. Jiang (JCT-B, 2001) showed that every graph $G$ with girth at least $2\ell+1$ and minimum degree at least $k/\ell$ contains every tree $T$ with $k$ edges whose maximum…

A digraph $D$ is $k$-linked if for any pair of two disjoint sets $\{x_{1},x_{2},\ldots,x_{k}\}$ and $\{y_{1},y_{2},\ldots,y_{k}\}$ of vertices in $D$, there exist vertex disjoint dipaths $P_{1},P_{2},\ldots,P_{k}$ such that $P_{i}$ is a…

Caccetta-H\"{a}ggkvist conjecture is a longstanding open problem on degree conditions that force an oriented graph to contain a directed cycle of a bounded length. Motivated by this conjecture, Kelly, K\"uhn, and Osthus initiated a study of…

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the…

For a given multigraph H, a graph G is H-linked, if |G| \geq |H| and for every injective map {\tau}: V (H) \rightarrow V (G), we can find internally disjoint paths in G, such that every edge from uv in H corresponds to a {\tau} (u) - {\tau}…

In 1960, Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if D is a directed graph on n vertices with minimum out-degree and in-degree at least n/2 (i.e. minimum semi-degree at least n/2), then D contains a directed…

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…

Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every $2$-connected non-bipartite graph with minimum degree at least $k+1$ contains $\lceil k/2\rceil $ cycles with consecutive odd lengths. In particular, they showed that this…

We ask the question, which oriented trees $T$ must be contained as subgraphs in every finite directed graph of sufficiently large minimum out-degree. We formulate the following simple condition: all vertices in $T$ of in-degree at least $2$…

Let $\delta$ and $\Delta$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of…

A well known theorem in graph theory states that every graph $G$ on $n$ vertices and minimum degree at least $d$ contains a path of length at least $d$, and if $G$ is connected and $n\ge 2d+1$ then $G$ contains a path of length at least…

Consider a directed multigraph $D$ that is balanced (i.e., at each vertex, the indegree equals the outdegree). Let $A$ be its set of arcs. Fix an integer $k$. Let $s$ be a vertex of $D$. We show that the number of $k$-element subsets $B$ of…

In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints, whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain…

We prove that if $D$ is a digraph of maximum outdegree and indegree at least $k$, and minimum semidegree at least $k/2$ that contains no oriented $4$-cycles, then $D$ contains each oriented tree $T$ with~$k$ arcs. This can be slightly…

Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. It is famous as one of the one hundred unsolved problems selected in…

Let $G$ be a finite, simple, and undirected graph of order $n$ and average degree $d$. Up to terms of smaller order, we characterize the minimal intervals $I$ containing $d$ that are guaranteed to contain some vertex degree. In particular,…

An $r$-uniform hypergraphic sequence (i.e., $r$-graphic sequence) $d=(d_1, d_2,\cdots,d_n)$ is said to be forcibly $k$-edge-connected if every realization of $d$ is $k$-edge-connected. In this paper, we obtain a strongest sufficient degree…