Related papers: Non-normal edge rings satisfying $(S_2)$-condition

Let $G$ be a $d$-regular graph on $n$ vertices. Frieze, Gould, Karo\'nski and Pfender began the study of the following random spanning subgraph model $H=H(G)$. Assign independently to each vertex $v$ of $G$ a uniform random number $x(v) \in…

We construct highly edge-connected $r$-regular graph which do not contain $r-2$ pairwise disjoint perfect matchings. The results partially answer a question stated by Thomassen [Factorizing regular graphs, J. Comb. Theory Ser. B (2019),…

We say a graph $G$ has a Hamiltonian path if it has a path containing all vertices of $G$. For a graph $G$, let $\sigma_2(G)$ denote the minimum degree sum of two nonadjacent vertices of $G$; restrictions on $\sigma_2(G)$ are known as…

In this paper, we compare the $\mathrm{v}$-numbers and the degree of the $h$-polynomials associated with edge ideals of connected graphs. We prove that the $\mathrm{v}$-number can be arbitrarily larger or smaller than the degree of the…

In 1972, Kotzig proved that for every even $n$, the complete graph $K_n$ can be decomposed into $\lceil\log_2n\rceil$ edge-disjoint regular bipartite spanning subgraphs, which is best possible. In this paper, we study regular bipartite…

Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of $G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain…

Jord\'an and Tanigawa recently introduced the $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G$. This is a quantitative measure of the $d$-dimensional rigidity of $G$ which generalizes the well-studied notion of spectral…

A graph $G$ is $k$-edge-Hamiltonian if any collection of vertex-disjoint paths with at most $k$ edges altogether belong to a Hamiltonian cycle in $G$. A graph $G$ is $k$-Hamiltonian if for all $S\subseteq V(G)$ with $|S|\le k$, the subgraph…

Let $ G=(V,E) $ be a simple graph of order $ n $ and size $ m $. A connected edge cover set of a graph is a subset $S$ of edges such that every vertex of the graph is incident to at least one edge of $S$ and the subgraph induced by $S$ is…

The cyclic edge-connectivity of a graph $G$ is the least $k$ such that there exists a set of $k$ edges whose removal disconnects $G$ into components where every component contains a cycle. We show that for graphs of minimum degree at least…

A graph is normal if it admits a clique cover $\mathcal C$ and a stable set cover $\mathcal S$ such that each clique in $\mathcal C$ and each stable set in $\mathcal S$ have a vertex in common. The pair $(\mathcal{C,S})$ is a normal cover…

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 $k$…

Let $G$ be a finite simple graph on the vertex set $V(G) = \{x_{1}, \ldots, x_{n}\}$ and match$(G)$, min-match$(G)$ and ind-match$(G)$ the matching number, minimum matching number and induced matching number of $G$, respectively. Let…

Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$ for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an $N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}. $$ If…

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\mathcal{G}(m, F)$ denote the family of $F$-free graphs with $m$ edges and without isolated vertices. Let $S_{n,k}$ denote the graph obtained by joining every vertex of…

In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly five or exactly three distinct colors, respectively. An edge is normal in an edge-coloring…

In an earlier work, finite groups whose power graphs are minimally edge connected have been classified. In this article, first we obtain a necessary and sufficient condition for an arbitrary graph to be minimally edge connected.…

Given a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-saturated if it is $\mathcal{F}$-free but the addition of any missing edge creates a copy of some $F \in \mathcal{F}$. The study of the minimum number of edges in…

The regular number of a graph G denoted by reg(G) is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is regular. In this work we answer to the problem posed as an…

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…