Related papers: Pseudo and Strongly Pseudo 2--Factor Isomorphic Re…

For a set $\mathcal{H}$ of connected graphs, a spanning subgraph $H$ of $G$ is called an $\mathcal{H}$-factor of $G$ if each component of $H$ is isomorphic to an element of $\mathcal{H}$. A graph $G$ is called an $\mathcal{H}$-factor…

In this study, we obtain the following two families of circulant graphs each has Type-2 isomorphic circulant graphs w.r.t. $m$ such that $m$ has more than one value. (i) Family of circulant graphs $C_{432}(R)$, each has isomorphic circulant…

This paper studies induced paths in strongly regular graphs. We give an elementary proof that a strongly regular graph contains a path $P_4$ as an induced subgraph if and only if it is primitive, i.e. it is neither a complete multipartite…

We show that there exist constants $\delta_1,\delta_2>0$ such that if $G$ is an $(n,d,\lambda)$-graph with $\lambda/d\le\delta_1$, then $G$ contains an induced cycle of length at least $\delta_2n/d$. We further demonstrate that, up to a…

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…

Let $F$ and $G$ be simple finite oriented graphs (without symmetric arcs). A graph $G$ is called $F$-irregular if any two distinct vertices in $G$ belong to a different number of subgraphs of $G$ isomorphic to $F$. In this paper, we…

A graph $X$ is said to be unstable if the direct product $X\times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is non-trivially unstable if it is…

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…

A graph with $v$ vertices is $(r)$-pancyclic if it contains precisely $r$ cycles of every length from 3 to $v$. A bipartite graph with even number of vertices $v$ is said to be $(r)$-bipancyclic if it contains precisely $r$ cycles of each…

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part…

A graph is $P_t$-free if it contains no induced subgraph isomorphic to a $t$-vertex path. A graph is not bipartite if and only if it contains an induced subgraph isomorphic to a $k$-vertex cycle, where $k$ is odd. We focus on the 3-coloring…

The $k$-dominating graph $D_k(G)$ of a graph $G$ is defined on the vertex set consisting of dominating sets of $G$ with cardinality at most $k$, two such sets being adjacent if they differ by either adding or deleting a single vertex. A…

We prove inequalities between the densities of various bipartite subgraphs in signed graphs and graphons. One of the main inequalities is that the density of any bipartite graph with girth r cannot exceed the density of the r-cycle. This…

A $k$-weak bisection of a cubic graph $G$ is a partition of the vertex-set of $G$ into two parts $V_1$ and $V_2$ of equal size, such that each connected component of the subgraph of $G$ induced by $V_i$ ($i=1,2$) is a tree of at most $k-2$…

A graph $\Gamma$ is said to be unstable if for the direct product $\Gamma \times K_2$, $Aut(\Gamma \times K_2)$ is not isomorphic to $Aut(\Gamma) \times \mathbb{Z}_2$. In this paper we show that a connected and non-bipartite Cayley graph…

Given a graph $G$ and a subset $X$ of vertices of $G$ with size at least two, we denote by $N^2_G(X)$ the set of vertices of $G$ that have at least two neighbors in $X$. We say that a bipartite graph $G$ with sides $A$ and $B$ satisfies the…

We study the flow spectrum ${\cal S}(G)$ and the integer flow spectrum $\overline{{\cal S}}(G)$ of signed $(2t+1)$-regular graphs. We show that if $r \in {\cal S}(G)$, then $r = 2+\frac{1}{t}$ or $r \geq 2 + \frac{2}{2t-1}$. Furthermore, $2…

Let $G$ be a graph, and $H\colon V(G)\to 2^\mathbb{N}$ a set function associated with $G$. A spanning subgraph $F$ of $G$ is called an $H$-factor if the degree of any vertex $v$ in $F$ belongs to the set $H(v)$. This paper contains two…

We classify all finite 2-groups that have a cyclic or dihedral maximal subgroup and determine their automorphism groups. Based on this result, we classify all pairs $ (G,\mathcal{M}) $, such that $ G $ is a finite 2-group and $ \mathcal{M}…

In 2002, D. Fon-Der-Flaass constructed a prolific family of strongly regular graphs. In this paper, we prove that for infinitely many natural numbers $n$, this family contains $n^{\Omega(n^{2/3})}$ strongly regular $n$-vertex graphs $X$…