Related papers: Turan problems for $k$-geodetic digraphs

Let $n \ge 5$ and $k\ge 4$ be positive integers. We determine the maximum size of digraphs of order n that avoid distinct walks of length k with the same endpoints. We also characterize the extremal digraphs attaining this maximum number…

Let $\overrightarrow{P_k}$ and $\overrightarrow{C_k}$ denote the directed path and the directed cycle of order $k$, respectively. In this paper, we determine the precise maximum size of $\overrightarrow{P_k}$-free digraphs of order $n$ as…

We suggest a new type of problem about distances in graphs and make several conjectures. As a first step towards proving them, we show that for sufficiently large values of n and k, a graph on n vertices that has no three vertices at…

Let $G$ be a connected graph and $\mathcal{P}(G)$ a graph parameter. We say that $\mathcal{P}(G)$ is feasible if $\mathcal{P}(G)$ satisfies the following properties: (I) $\mathcal{P}(G)\leq \mathcal{P}(G_{uv})$, if $G_{uv}=G[u\to v]$ for…

For integers $k, \ell \geq 3$, let $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ denote the maximum number of directed cycles of length $k$ in any oriented graph on $n$ vertices which does not contain a directed cycle of…

A graph $G$ is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. We…

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…

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph…

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…

A graph $G$ is $k$-edge geodetic graph if every edge of $G$ lies in at least one geodesic of length $k$. We studied some basic properties of $k$-edge geodetic graphs. We investigated the $k$ edge-geodeticity of complete bipartite graph…

The Tur\'an problem asks for the largest number of edges in an $n$-vertex graph not containing a fixed forbidden subgraph $F$. We construct a new family of graphs not containing $K_{s,t}$, for $t= C^s$, with $\Omega(n^{2-1/s})$ edges…

We prove an asymptotic result on the maximum number of k-vertex subtrees in binary trees of given order. This problem turns out to be equivalent to determine the maximum number of k+2-cycles in n-vertex outerplanar graphs, thus we settle…

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…

In a generalized Tur\'an problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $F$-free graph of order $n$. We consider generalized Tur\'an problems where the host graph is planar. In particular we…

Let $n,k,t$ be positive integers. What is the maximum number of arcs in a digraph on $n$ vertices in which there are at most $t$ distinct walks of length $k$ with the same endpoints? In this paper, we prove that the maximum number is equal…

For a positive integer $k\ge 1$, a graph $G$ is $k$-stepwise irregular ($k$-SI graph) if the degrees of every pair of adjacent vertices differ by exactly $k$. Such graphs are necessarily bipartite. Using graph products it is demonstrated…

Two vertices $u, v \in V$ of an undirected connected graph $G=(V,E)$ are resolved by a vertex $w$ if the distance between $u$ and $w$ and the distance between $v$ and $w$ are different. A set $R \subseteq V$ of vertices is a $k$-resolving…

A classical Tur\'an problem asks for the maximum possible number of edges in a graph of a given order that does not contain a particular graph $H$ as a subgraph. It is well-known that the chromatic number of $H$ is the graph parameter which…

A vertex set $S$ of a graph $G$ is geodetic if every vertex of $G$ lies on a shortest path between two vertices in $S$. Given a graph $G$ and $k \in \mathbb N$, the NP-hard Geodetic Set problem asks whether there is a geodetic set of size…

Given a graph $G = (V,E)$, a set $S \subset V$ is called a $k$-\emph{metric generator} for $G$ if any pair of different vertices of $G$ is distinguished by at least $k$ elements of $S$. A graph is $k$-\emph{metric dimensional} if $k$ is the…