Related papers: Locally connected graphs: metric properties

We study the Localization game on locally finite graphs trees, where each of the countably many vertices have finite degree. In contrast to the finite case, we construct a locally finite tree with localization number $n$ for any choice of…

The metric dimension of non-component graph, associated to a finite vector space, is determined. It is proved that the exchange property holds for resolving sets of the graph, except a special case. Some results are also related to an…

A connected graph is called \emph{geodetic} if there is a unique geodesic between each pair of vertices. In this paper we prove that if a finitely generated group admits a Cayley graph which is geodetic, then the group must be virtually…

We find the asymptotic number of connected graphs with $k$ vertices and $k-1+l$ edges when $k,l$ approach infinity, reproving a result of Bender, Canfield and McKay. We use the {\em probabilistic method}, analyzing breadth-first search on…

Following a general program of studying limits of discrete structures, and motivated by the theory of limit objects of converge sequences of dense simple graphs, we study the limit of graph sequences such that every edge is labeled by an…

We determine an asymptotic formula for the number of labelled 2-connected (simple) graphs on $n$ vertices and $m$ edges, provided that $m-n\to\infty$ and $m=O(n\log n)$ as $n\to\infty$. This is the entire range of $m$ not covered by…

Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable…

Given a finite set $E$, a subset $D\sub E$ (viewed as a function $E\to \F_2$) is orthogonal to a given subspace $\FF$ of the $\F_2$-vector space of functions $E\to \F_2$ as soon as $D$ is orthogonal to every $\sub$-minimal element of $\FF$.…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

We first introduce a notion of convex structure in generalized metric spaces, then we introduce tripartite contractions, tripartite semi-contractions, tripartite coincidence points, as well as tripartite best proximity points for a given…

A graph $G$ is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. It is well known that a 2-connected, $k$-regular graph $G$ on at most $3k-1$ vertices is edge-Hamiltonian if for every edge…

A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was…

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected…

For any pair of edges $e,f$ of a graph $G$, we say that {\em $e,f$ are $P_3$-connected in $G$} if there exists a sequence of edges $e=e_0,e_1,\ldots, e_k=f$ such that $e_i$ and $e_{i+1}$ are two edges of an induced $3$-vertex path in $G$…

We derive an asymptotic formula for the number of strongly connected digraphs with $n$ vertices and $m$ arcs (directed edges), valid for $m-n\to\infty$ as $n\to \infty$ provided $m=O(n\log n)$. This fills the gap between Wright's results…

In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph $E$ is strongly connected and contains a finite amount of vertices then a locally compact…

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is \textit{$k$-linked} if,…

The center, median and the security center are three central parts defined for any connected graph whereas the characteristic set, subtree core and core vertices are three central parts defined for trees only. We extend the concept of the…

From the point of view of discrete geometry, the class of locally finite transitive graphs is a wide and important one. The subclass of Cayley graphs is of particular interest, as testifies the development of geometric group theory. Recall…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, a path in $G$ is said to be an $S$-path if it connects all vertices of $S$. Two $S$-paths $P_1$ and $P_2$ are said to be internally disjoint if $E(P_1)\cap…