Related papers: Highly edge-connected regular graphs without large…

In this paper, we show that for any positive integer $m$ and $k\in [2]$, let $G$ be a $(2m+2k+2)$-connected graph and let $a_1,\ldots , a_m, s, t$ be any distinct vertices of $G$, there are $k$ internally disjoint $s$-$t$ paths $P_1,…

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. If every quasi $k$-connected graph without a quasi $k$-contractible edge has either $H_{1}$…

An old problem raised independently by Jacobson and Sch\"onheim asks to determine the maximum $s$ for which every graph with $m$ edges contains a pair of edge-disjoint isomorphic subgraphs with $s$ edges. In this paper we determine this…

We determine the structure of automorphism groups of finite graphs of bounded Hadwiger number. Our proof includes a structural analysis of finite edge-transitive graphs. In particular, we show that for connected, $K_{h+1}$-minor-free,…

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$…

An $r$-graph is an $r$-regular graph with no odd cut of size less than $r$. A well-celebrated result due to Lov\'asz says that for such graphs the linear system $Ax = \textbf{1}$ has a solution in $\mathbb{Z}/2$, where $A$ is the $0,1$ edge…

Considering regular graphs with every edge in a triangle we prove lower bounds for the number of triangles in such graphs. For r-regular graphs with r <= 5 we exhibit families of graphs with exactly that number of triangles and then…

In this paper, we introduce a corresponding between bipartite graphs with a perfect matching and digraphs, which implicates an equivalent relation between the extendibility of bipartite graphs and the strongly connectivity of digraphs. Such…

In this article, we discuss when one can extend an r-regular graph to an r + 1 regular by adding edges. Different conditions on the num- ber of vertices n and regularity r are developed. We derive an upper bound of r, depending on n, for…

For each $b \geq 5$ we construct a pair of cospectral $b$-regular graphs, where one has a perfect matching and the other one not. This solves a research problem posed by the third author at the 22nd British Combinatorial Conference.

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.…

An {\em ordered $r$-graph} is an $r$-uniform hypergraph whose vertex set is linearly ordered. Given $2\leq k\leq r$, an ordered $r$-graph $H$ is {\em interval} $k$-{\em partite} if there exist at least $k$ disjoint intervals in the ordering…

Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th…

A connected matching in a graph G consists of a set of pairwise disjoint edges whose covered vertices induce a connected subgraph of G. While finding a connected matching of maximum cardinality is a well-solved problem, it is NP-hard to…

The twisted graph $T_{n}$ is a drawing of the complete graph with $n$ vertices $v_{1},v_{2},\ldots ,v_{n}$ in which two edges $v_{i}v_{j}$ ($i<j$) and $v_{s}v_{t}$ ($s<t$) cross if and only if $i<s<t<j$ or $s<i<j<t$. We show that for any…

Edge polytopes is a class of interesting polytope with rich algebraic and combinatorial properties, which was introduced by Ohsugi and Hibi. In this papar, we follow a previous study on cutting edge polytopes by Hibi, Li and Zhang. Instead…

A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$, and a graph $G$ is perfectly weight divisible if for every…

We investigate graphs that have characteristic-dependent well-covered dimension and show how more of these graphs can be constructed from known ones.

A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a…

We study rotation $r$-graphs and show that for every $r$-graph $G$ of odd regularity there is a simple rotation $r$-graph $G'$ such that $G$ can be obtained form $G'$ by a finite number of $2$-cut reductions. As a consequence, some hard…