Related papers: On Cyclic Kautz Digraphs

Let $k_r(n,\delta)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $\delta$. We evaluate $k_r(n,\delta)$ for $\delta \leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture…

A set $S$ of vertices of a digraph $D$ is called an open neighbourhood locating-dominating set if every vertex in $D$ has an in-neighbour in $S$, and for every pair $u,v$ of vertices of $D$, there is a vertex in $S$ that is an in-neighbour…

A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there…

A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of…

In this paper, we first give a new result characterizing the strongly connected digraphs with a diameter equal to that of their line digraphs. Then, we introduce the concepts of the inner diameter and inner radius of a digraph and study…

Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by…

Let $\ell$ denote a positive integer. A connected graph $\G$ of diameter at least $\ell$ is said to be $\ell${\it -distance-balanced} whenever for any pair of vertices $u,v$ of $\G$ such that $d(u,v)=\ell$, the number of vertices closer to…

This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected…

In recent years there has been increased interest in extremal problems for "counting" parameters of graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. In the same…

Let $H$ be a graph with maximum degree $d$, and let $d'\ge 0$. We show that for some $c>0$ depending on $H,d'$, and all integers $n\ge 0$, there are at most $c^n$ unlabelled simple $d$-connected $n$-vertex graphs with maximum degree at most…

A graph is $k$-chordal if it does not have an induced cycle with length greater than $k$. We call a graph chordal if it is $3$-chordal. Let $G$ be a graph. The distance between the vertices $x$ and $y$, denoted by $d_{G}(x,y)$, is the…

K-geodetic graphs (K capital) are defined as graphs in which each pair of nonadjacent vertices has at most K paths of minimum length between them. A K-geodetic graph is geodetic if K=1, bigeodetic if K=2 and trigeodetic if K=3. K-geodetic…

A graph is called $k$-critical if its chromatic number is $k$ but any proper subgraph has chromatic number less than $k$. An old and important problem in graph theory asks to determine the maximum number of edges in an $n$-vertex…

Motivated by Kuratowski's theorem, a Kuratowski subgraph of a graph is a subgraph that is a subdivided $K_5$ or a subdivided $K_{3,3}$. An edge is crossing-critical if the crossing number decreases after removing the edge. In this note, we…

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(\delta\geq k+1)}$. Let $m=cn$ and…

In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$,…

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph…

Characterization of k-chordal graphs based on the existence of a "simplicial path" was shown in [Chv{\'a}tal et al. Note: Dirac-type characterizations of graphs without long chordless cycles. Discrete Mathematics, 256, 445-448, 2002]. We…

Let $ H $ be a multi-digraph on $ h $ vertices with $ q $ arcs. An \textbf{$H$-subdivision} in a digraph $D$ is a subdigraph obtained by replacing every arc $uv$ of $H$ with a path from $u$ to $v$ in $D$ such that these paths are pairwise…

A non-empty set $X$ of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by $X$ is connected and it is called convex if no two vertices of $X$ are connected by a directed path in which some…