Related papers: Spanning Simplicial Ccomplexes of Uni-Cyclic Graph…

A graph can be regarded as an electrical network in which each edge is a resistor. This point of view relates combinatorial quantities, such as the number of spanning trees, to electrical ones such as effective resistance. The second and…

We consider a uniform spanning tree in a $\delta$-square grid approximation of a planar domain $\Omega$. For given integer $n\ge 2$, we condition the tree on the following $n$-arm event: we pick $n$ branches, emanating from $n$ points…

We give a new proof of the Skeletal Lemma, which is the main technical tool in our paper on Hamilton cycles in line graphs [T. Kaiser and P. Vr\'ana, Hamilton cycles in 5-connected line graphs, European J. Combin. 33 (2012), 924-947]. It…

Generalizing pseudospherical drawings, we introduce a new class of simple drawings, which we call separable drawings. In a separable drawing, every edge can be closed to a simple curve that intersects each other edge at most once. Curves of…

The Kneser Graph $K(n,k)$ has as vertices all $k$-subsets of $\{1,\ldots,n\}$ and edges connecting two vertices if they are disjoint. The $s$-stable Kneser Graph $K_{s-stab}(n, k)$ is obtained from the Kneser graph by deleting vertices with…

Let $G$ be a finite group. Define a graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to be adjacent if and only if $\langle x,y\rangle$ is cyclic. Denote this graph by $\Delta(G)$. The graph…

The complexity of a graph is the number of its labeled spanning trees. It is demonstrated that the seven known triangle-free strongly regular graphs, such as the Higman-Sims graph, are graphs of maximal complexity among all graphs of the…

Let $H$ be a simple undirected graph and $G=\mathrm{L}(H)$ be its line graph. Assume that $\Delta(G)$ denotes the clique complex of $G$. We show that $\Delta(G)$ is sequentially Cohen-Macaulay if and only if it is shellable if and only if…

We prove a topological extension of Dirac's theorem suggested by Gowers in 2005: for any connected, closed surface $\mathscr{S}$, we show that any two-dimensional simplicial complex on $n$ vertices in which each pair of vertices belongs to…

In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph…

A graph is odd if all of its vertices have odd degrees. In particular, an odd spanning tree in a connected graph is a spanning tree in which all vertices have odd degrees. In this paper we establish a unified technique to enumerate odd…

In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, $R-$graph, central graph. Also, it generalizes the construction of overlay graph (Marius Somodi \emph{et al.}, 2017), and…

Let $k\geq2$ be an integer. A $k$-tree is a tree with maximum degree at most $k$. In this paper, we give a closure result on spanning $k$-trees of graphs with given minimum degree. Let $\delta\geq1$ be an integer, and $G$ be a connected…

The theory of $k$-regular graphs is closely related to group theory. Every $k$-regular, bipartite graph is a Schreier graph with respect to some group $G$, a set of generators $S$ (depending only on $k$) and a subgroup $H$. The goal of this…

Let ${\mathcal U}_{\delta}$ be the uniform spanning tree on $\delta \mathbb{Z}^{3}$. A spanning cluster of ${\mathcal U}_{\delta}$ is a connected component of the restriction of ${\mathcal U}_{\delta}$ to the unit cube $[0,1]^{3}$ that…

The (k,d)-hypersimplex is a (d-1)-dimensional polytope whose vertices are the (0,1)-vectors that sum to k. When k=1, we get a simplex whose graph is the complete graph with d vertices. Here we show how many of the well known graph…

We study dismantlability in graphs. In order to compare this notion to similar operations in posets (partially ordered sets) or in simplicial complexes, we prove that a graph G dismants on a subgraph H if and only if H is a strong…

Motivated to find the answers to some of the questions that have occurred in recent papers dealing with Hamiltonian cycles (abbreviated HCs) in some special classes of grid graphs we started the investigation of spanning unions of cycles,…

We extend the edge version of the classical Menger's Theorem for undirected graphs to $n$-dimensional simplicial complexes with chains over the field $\mathbb{F}_2$. The classical Menger's Theorem states that two different vertices in an…

A simplicial complex $\Delta$ is a virtually Cohen-Macaulay simplicial complex if its associated Stanley-Reisner ring $S$ has a virtual resolution, as defined by Berkesch, Erman, and Smith, of length ${\rm codim}(S)$. We provide a…