Related papers: Digraphs with degree two and excess two are diregu…
An important topic in the design of efficient networks is the construction of $(d,k,+\epsilon )$-digraphs, i.e. $k$-geodetic digraphs with minimum out-degree $\geq d$ and order $M(d,k)+ \epsilon $, where $M(d,k)$ represents the Moore bound…
Moore digraphs, that is digraphs with out-degree $d$, diameter $k$ and order equal to the Moore bound $M(d,k) = 1 + d + d^2 + \dots +d^k$, arise in the study of optimal network topologies. In an attempt to find digraphs with a `Moore-like'…
A digraph $G$ is \emph{$k$-geodetic} if for any (not necessarily distinct) vertices $u,v$ there is at most one directed walk from $u$ to $v$ with length not exceeding $k$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is…
A digraph $G$ is \emph{$k$-geodetic} if for any pair $u,v \in V(G)$ there is at most one $u,v$-walk of length not exceeding $k$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the directed Moore bound…
The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore…
An almost Moore digraph is a diregular digraph of degree $d>1$, diameter $k>1$ and order $d+d^2+ \cdots +d^k$. Their existence has only been shown for $k=2$. It has also been conjectured that there are no more almost Moore digraphs, but so…
In this paper we consider the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of vertices in a graph of maximum degree {\Delta} and diameter D. In this context, the…
It is known that the number of vertices of a graph of diameter two cannot exceed $d^2+1$. In this contribution we give a new lower bound for orders of Cayley graphs of diameter two in the form $C(d,2)>0.684d^2$ valid for all degrees $d\geq…
The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of…
We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree $d$. We completely determine the asymptotic behaviour…
Mader conjectured that for all k there is an integer d(k) such that every digraph of minimum outdegree at least d(k) contains a subdivision of a transitive tournament of order k. In this note we observe that if the minimum outdegree of a…
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…
The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum…
The degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the \emph{degree/geodecity} problem concerns the smallest order of a $k$-geodetic mixed…
A graph is diameter-$k$-critical if its diameter equals $k$ and the deletion of any edge increases its diameter. The Murty-Simon Conjecture states that for any diameter-2-critical graph $G$ of order $n$, $e(G) \leq \lfloor…
We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$…
We analyse an extremal question on the degrees of the link graphs of a finite regular graph, that is, the subgraphs induced by non-trivial spheres. We show that if $G$ is $d$-regular and connected but not complete then some link graph of…
A special case of a conjecture by Thomass\'e is that any oriented graph with minimum outdegree k contains a dipath of length 2k. For the sake of proving whether or not a counterexample exists, we present reductions and establish bounds on…
Let $C_{d,k}$ be the largest number of vertices in a Cayley digraph of degree $d$ and diameter $k$, and let $BC_{d,k}$ be the largest order of a bipartite Cayley digraph for given $d$ and $k$. For every degree $d\geq2$ and for every odd $k$…
Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta< 2\sqrt{k-1}$. Our…