Related papers: Double-Star Decomposition of Regular Graphs

In this paper we consider $r$-regular graphs $G$ that admit the vertex set partition such that one of the induced subgraphs is the join of an $s$-vertex clique and a $t$-vertex co-clique and represents a star complement for an eigenvalue…

A Graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their complements. Then we split characterizing…

Let $k\geq\ell\geq1$ and $n\geq 1$ be integers. Let $G(k,n)$ be the complete $k$-partite graph with $n$ vertices in each colour class. An $\ell$-decomposition of $G(k,n)$ is a set $X$ of copies of $K_k$ in $G(k,n)$ such that each copy of…

The $k$-coalescence of two graphs is obtained by merging a $k$-clique of each graph. The $A_\alpha$-matrix of a graph is the convex combination of its degree matrix and adjacency matrix. In this paper, we present some structural properties…

Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $\alpha_1 e(H_1) + ... + \alpha_k e(H_k) = m$ where each $\alpha_i$ is a…

The Tur\'an number of a graph $F$, $ex(n,F)$, is the maximum number of edges in a graph on $n$ vertices which does not contain $F$ as a subgraph. Let $S_{a,b}$ denote a double star with a central edge $uv$, $a$ leaves connected to $u$ and…

A $k$-star-forest is a forest with at most $k$ connected components where each component is a star. Let $F_k(n)$ be the minimum integer such that the complete graph on $n$ vertices can be decomposed into $F_k(n)$ $k$-star-forests. Pach,…

Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…

Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of all complex irreducible characters of $G$. The character-graph $\Delta(G)$ associated to $G$, is a graph whose vertex set is the set of primes which divide the degrees of some…

A set $D$ of vertices of a graph $G$ is a dissociation set if each vertex of $D$ has at most one neighbor in $D$. The dissociation number of $G$, $diss(G)$, is the cardinality of a maximum dissociation set in a graph $G$. In this paper we…

A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let…

A \emph{proportionally dense subgraph} (PDS) is an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the rest of the graph. In this paper, we study…

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A subset $I$ of $V(G)$ is an independent vertex subset if no two vertices in $I$ are adjacent in $G$. We study the number, $\sigma_1(G)$, of all subsets of $v(G)$ that contain…

An edge (vertex) cut $X$ of $G$ is $r$-essential if $G-X$ has two components each of which has at least $r$ edges. A graph $G$ is $r$-essentially $k$-edge-connected (resp. $k$-connected) if it has no $r$-essential edge (resp. vertex) cuts…

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions…

We say that a graph G is $(k,\ell)$-stable if removing $k$ vertices from it reduces its independence number by at most $\ell$. We say that G is tight $(k,\ell)$-stable if it is $(k,\ell)$-stable and its independence number equals…

A $k$-edge-coloured graph is colour-balanced if each colour appears equally often. Resolving a conjecture of Pardey and Rautenbach, we show that any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a perfect matching that…

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation…

Given a tree T on v vertices and an integer k exceeding one. One can define the k-expansion T^k as a k-uniform linear hypergraph by enlarging each edge with a new, distinct set of (k-2) vertices. Then T^k has v+ (v-1)(k-2) vertices. The aim…