Related papers: On Cyclic Kautz Digraphs

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is…

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…

In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number…

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…

We introduce a new notion of circular colourings for digraphs. The idea of this quantity, called star dichromatic number $\vec{\chi}^\ast(D)$ of a digraph $D$, is to allow a finer subdivision of digraphs with the same dichromatic number…

A Helly circular-arc model M = (C,A) is a circle C together with a Helly family \A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly…

Dean conjectured that for each integer $k \ge 3$, every graph with minimum degree at least $k$ has a cycle whose length is divisible by $k$; this conjecture is known to be true for all $k\neq 5$. For $k\in\{3,4\}$, stronger statements are…

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have…

The subgraph number of a vertex in a graph is defined as the number of connected subgraphs containing that vertex. The graph and its vertex which correspond to the minimum subgraph number among all graphs on $n$ vertices and $k$ cut…

An $L(d,1)$-labeling of a graph $G$ is an assignment of nonnegative integers to the vertices such that adjacent vertices receive labels that differ by at least $d$ and those at a distance of two receive labels that differ by at least one,…

A coloring of a digraph is a partition of its vertex set such that each class induces a digraph with no directed cycles. A digraph is $k$-chromatic if $k$ is the minimum number of classes in such partition, and a digraph is oriented if…

Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$.…

We prove that the clique graph operator $k$ is divergent on a locally cyclic graph $G$ (i.e. $N_G(v)$ is a circle) with minimum degree $\delta(G)=6$ if and only if $G$ is $6$-regular. The clique graph $kG$ of a graph $G$ has the maximal…

A classical result by Erd\H{o}s, and later on by Bondy and Simonivits, states that every $n$-vertex graph with no cycle of length $2k$ has at most $O(n^{1+1 /k})$ edges. This bound is known to be tight when $k \in \{2,3,5\},$ but it is a…

Given a digraph $D$, let $c(D)$ denote the largest integer $k$ such that there are $k$ openly disjoint cycles through a vertex, i.e., a collection of directed cycles $C_1,\ldots,C_k$ through a common vertex $v$ such that…

For $t \ge 2$, let us call a digraph $D$ \emph{t-chordal} if all induced directed cycles in $D$ have length equal to $t$. In a previous paper, we asked for which $t$ it is true that $t$-chordal graphs with bounded clique number have bounded…

Let $D$ be a digraph. We define the minimum semi-degree of $D$ as $\delta^{0}(D) := \min \{\delta^{+}(D), \delta^{-}(D)\}$. Let $k$ be a positive integer, and let $S = \{s\}$ and $T = \{t_{1}, \dots ,t_{k}\}$ be any two disjoint subsets of…

A graph with vertex set V and edge set E is called a (d,c)-expander if the maximum degree of a vertex is d and, for every subset W of V that has cardinality at most |V|/2, the number of edges between vertices in W and vertices outside of W…

Let $k$ be an integer with $k\geq 2$. A $k$-king in a digraph $D$ is a vertex which can reach every other vertex by a directed path of length at most $k$ and a non-king is a vertex which is not a 3-king. A subset $K$ is $k$-independent if…

We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andr\'asfai, Erd\H os, and S\'os implies that every $n$-vertex graph with odd girth $2k+1$ and minimum…