Related papers: On conjectures and problems of Ruzsa concerning di…

We prove the following 30-year old conjecture of Gy\H{o}ri and Tuza: the edges of every $n$-vertex graph $G$ can be decomposed into complete graphs $C_1,\ldots,C_\ell$ of orders two and three such that $|C_1|+\cdots+|C_\ell|\le…

Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this…

Let $G$ be a finite group. For some fixed prime $p$, let $\Gamma_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of…

The non--commuting graph $\Gamma(G)$ of a non--abelian group $G$ is defined as follows. The vertex set $V(\Gamma(G))$ of $\Gamma(G)$ is $G\setminus Z(G)$ where $Z(G)$ denotes the center of $G$ and two vertices $x$ and $y$ are adjacent if…

Let $G$ be a finite group. The solubility graph associated with the finite group $G$, denoted by $\Gamma_{\cal S}(G)$, is a simple graph whose vertices are the non-trivial elements of $G$, and there is an edge between two distinct elements…

Given a finite simple undirected graph $G$, let $T_1(G)$ denote the subset of vertices of $G$ such that every vertex of $T_1(G)$ belongs to at least one subgraph isomorphic to a graph obtained by connecting a single vertex to two vertices…

For a graph $G$, let $f(G)$ be the largest integer $k$ for which there exist two vertex-disjoint induced subgraphs of $G$ each on $k$ vertices, both inducing the same number of edges. We prove that $f(G) \ge n/2 - o(n)$ for every graph $G$…

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 $H$…

In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We…

For every graph $X$, we consider the class of all connected $\{K_{1,3}, X\}$-free graphs which are distinct from an odd cycle and have independence number at least $4$, and we show that all graphs in the class are perfect if and only if $X$…

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$…

The vertex connectivity of a graph $G$ is the size of the smallest set of vertices $S$ such that $G \setminus S$ is disconnected. For the class of planar graphs, the problem of vertex connectivity is well-studied, both from structural and…

Let G be a finite graph on [n] = {1,2,3,...,n}, X a 2 times n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this paper, we study about ideals I_G of S generated by 2-minors [i,j] of X which correspond to…

An independent vertex subset $S$ of the directed graph $G$ is a kernel if the set of out-neighbors of $S$ is $V(G)\setminus S$. An independent vertex subset $Q$ of $G$ is a quasi-kernel if the union of the first and second out-neighbors…

The graph reconstruction conjecture states that all graphs on at least three vertices are determined up to isomorphism by their deck. In this paper, a general framework for this problem is proposed to simply explain the reconstruction of…

The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard…

Chudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs [1]. In the present paper we reprove their theorem using many of the same ideas, but with the key…

Graphlets are subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence (gds), gives a topological description of the surrounding of the analyzed vertex.…

Let G be a finite group. Denoting by cd(G) the set of degrees of the irreducible complex characters of G, we consider the character degree graph of G: this is the (simple undirected) graph whose vertices are the prime divisors of the…

Let $G$ be a $2$-generated group. The generating graph $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g_1$ and $g_2$ are adjacent if $G = \langle g_1, g_2 \rangle.$ This graph encodes the…