Related papers: The linkedness of cubical polytopes: The cube

A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…

A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected…

Let $D$ be a strongly connected digraphs on $n\ge 4$ vertices. A vertex $v$ of $D$ is noncritical, if the digraph $D-v$ is strongly connected. We prove, that if sum of the degrees of any two adjacent vertices of $D$ is at least $n+1$, then…

For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected…

A k-tree is either a complete graph on (k+1) vertices or given a k-tree G' with n vertices, a k-tree G with (n+1) vertices can be constructed by introducing a new vertex v and picking a k-clique Q in G' and then joining each vertex u in Q.…

A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the…

We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.

Let $G$ be a graph of order $n$ and let $u,v$ be vertices of $G$. Let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then the average connectivity $\overline{\kappa}(G)$ of $G$, is defined as $…

Let $C_{s,t}$ be the complete bipartite geometric graph, with $s$ and $t$ vertices on two distinct parallel lines respectively, and all $s t$ straight-line edges drawn between them. In this paper, we show that every complete bipartite…

Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. Shi and Shan conjectured that every $1$-tough $2k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian for $k \geq 4$. This conjecture has been…

In this paper we show that graphs of "neighbourly" cubical complexes -- cubical complexes in which every pair of vertices spans a (unique) cube -- have good expansion properties, using a technique based on multicommodity flows. By showing…

Take any $d + 3$ points in $\mathbb{R}^d$. It is known that (a) if $d = 2k + 1$, then there are two linked $(k + 1)$-simplices with the vertices at these points; (b) if $d = 2k$, then there are two disjoint $(k + 1)$-tuples of these points…

A graph $G$ is equimatchable if any matching in $G$ is a subset of a maximum-size matching. It is known that any $2$-connected equimatchable graph is either bipartite or factor-critical. We prove that for any vertex $v$ of a $2$-connected…

In this paper, we show that every $2m$-partition-connected graph $G$ has a bipartite $m$-partition-connected factor $H$ such that for each vertex $v$, $d_H(v)\le \lceil \frac{3}{4}d_G(v)\rceil$. A graph $H$ is said to be…

A $d$-dimensional framework is a pair $(G,p)$, where $G$ is a graph and $p$ maps the vertices of $G$ to points in $\mathbb{R}^d$. The edges of $G$ are mapped to the corresponding line segments. A graph $G$ is said to be globally rigid in…

Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$…

Let $Q_k$ denote the $k$-dimensional hypercube on $2^k$ vertices. A vertex in a subgraph of $Q_k$ is {\em full} if its degree is $k$. We apply the Kruskal-Katona Theorem to compute the maximum number of full vertices an induced subgraph on…

Let v(G) be the number of vertices and t(G,k) the maximum number of disjoint k-edge trees in G. In this paper we show that (a1) if G is a graph with every vertex of degree at least two and at most s, where s > 3, then t(G,2) is at least…

In [{Structural properties and decomposition of linear balanced matrices}, {\it Mathematical Programming}, 55:129--168, 1992], Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of…

It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…