Related papers: Long-eared digraphs

A digraph $D=(V,A)$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in\{3,...,n\}$; and D is vertex-pancyclic iff, for each vertex $v\in V$ and each $k\in \{3,...,n\}$, $D$ contains a…

It is well known that determining if a digraph has a kernel is an NP-complete problem. However, Topp proved that when subdividing every arc of a digraph we obtain a digraph with a kernel. In this paper we define the kernel subdivision…

Let $H$ be a digraph possibly with loops and $D$ a digraph without loops with a coloring of its arcs $c:A(D) \rightarrow V(H)$ ($D$ is said to be an $H$-colored digraph). A directed path $W$ in $D$ is said to be an $H$-path if and only if…

Let $H = (V_H, A_H)$ be a digraph which may contain loops, and let $D = (V_D, A_D)$ be a loopless digraph with a coloring of its arcs $c: A_D \to V_H$. An $H$-walk of $D$ is a walk $(v_0, \dots, v_n)$ of $D$ such that $(c(v_{i-1}, v_i),…

A graph $H$ is an \emph{isometric} subgraph of $G$ if $d_H(u,v)= d_G(u,v)$, for every pair~$u,v\in V(H)$. A graph is \emph{distance preserving} if it has an isometric subgraph of every possible order. A graph is \emph{sequentially distance…

We study $k$-colored kernels in $m$-colored digraphs. An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K$ of its vertices such that (i) from every vertex $v\notin K$ there exists an at most $k$-colored directed…

The dichromatic number $\vec\chi(D)$ of a digraph is the minimum number $k$ such that $V(D)$ can be partitioned into $k$ subsets, each inducing an acyclic digraph. The acyclic number $\vec\alpha(D)$ is the cardinality of a largest induced…

We define and study a new structural parameter for directed graphs, which we call \emph{ear anonymity}. Our parameter aims to generalize the useful properties of \emph{funnels} to larger digraph classes. In particular, funnels are exactly…

We prove that for every oriented graph $D$ and every choice of positive integers $k$ and $\ell$, there exists an oriented graph $D^*$ along with a surjective homomorphism $\psi\colon V(D^*) \to V(D)$ such that: (i) girth$(D^*) \geq\ell$;…

An acyclic digraph each vertex of which has indegree at most $i$ and outdegree at most $j$ is called an $(i, j)$ digraph for some positive integers $i$ and $j$. Lee {\it et al.} (2017) studied the phylogeny graphs of $(2, 2)$ digraphs and…

Let $H=(V_H,A_H)$ be a digraph, possibly with loops, and let $D=(V_D, A_D)$ be a loopless multidigraph with a colouring of its arcs $c: A_D \rightarrow V_H$. An $H$-path of $D$ is a path $(v_0, \dots, v_n)$ of $D$ such that $(c(v_{i-1},…

Let $T$ be a digraph with vertices $u_1, \dots, u_t$ ($t\ge 2$) and let $H_1, \dots, H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1, \dots, H_t]$ is a digraph with vertex set…

In a strong edge-coloring of a graph $G=(V,E)$, any two edges of distance at most $2$ get distinct colors. The strong chromatic index of $G$, denoted by $\chi_s'(G)$, is the minimum number of colors needed in a strong edge-coloring of $G$.…

Mader [J. Graph Theory 65 (2010) 61-69] conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree…

A $(k_1+k_2)$-bispindle is the union of $k_1$ $(x,y)$-dipaths and $k_2$ $(y,x)$-dipaths, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. showed that for every $(1,1)$- bispindle $B$, there exists an integer $k$…

The chromatic profile of a digraph $H$, denoted by $\delta_{\chi}^{+}(H,k)$, is the infimum $d$ such that any $H$-free digraph $D$ on $n$ vertices with minimum out-degree $\delta^{+}(D) \ge dn$ must be $k$-colorable. We determine the exact…

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…

A digraph is strongly connected if it has a directed path from $x$ to $y$ for every ordered pair of distinct vertices $x, y$ and it is strongly $k$-connected if it has at least $k+1$ vertices and remains strongly connected when we delete…

A {\em quasi-kernel} of a digraph $D$ is an independent set $Q\subseteq V(D)$ such that for every vertex $v\in V(D)\backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $u\in Q$. In 1974, Chv\'{a}tal and…

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph $D$ is $k$-dicritical if $\vec{\chi}(D)…