Related papers: An extremal problem on potentially $K_{p_{1},p_{2}…

The tree spanner problem for a graph $G$ is as follows: For a given integer $k$, is there a spanning tree $T$ of $G$ (called a tree $k$-spanner) such that the distance in $T$ between every pair of vertices is at most $k$ times their…

A permutation $\sigma\in S_n$ is said to be $k$-universal or a $k$-superpattern if for every $\pi\in S_k$, there is a subsequence of $\sigma$ that is order-isomorphic to $\pi$. A simple counting argument shows that $\sigma$ can be a…

Let $n,r,k,s$ be positive integers with $n,k\ge 2$. The generalized Ramsey number $R(n,r;k,s)$ is the smallest positive integer $p$ such that for every graph $G$ of order $p$, either $G$ contains a subgraph induced by $n$ vertices with at…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. $1$- and $2$-factor-critical graphs are the well-known factor-critical and…

Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\ldots,k$. Then $k\geq \Delta(G)$, while by Vizing's theorem, no more than $k=\Delta(G)+1$ is necessary for constructing such $c$. On the course of investigating…

Let v(G) be the number of vertices and t(G,k) the maximum number of disjoint k-edge trees in G. In this paper we show that (a1) if G is a graph with every vertex of degree at least two and at most s, where s > 3, then t(G,2) is at least…

Given a graph $G$, the optimization version of the graph burning problem seeks for a sequence of vertices, $(u_1,u_2,...,u_p) \in V(G)^p$, with minimum $p$ and such that every $v \in V(G)$ has distance at most $p-i$ to some vertex $u_i$.…

We study the model $G_\alpha\cup G(n,p)$ of randomly perturbed dense graphs, where $G_\alpha$ is any $n$-vertex graph with minimum degree at least $\alpha n$ and $G(n,p)$ is the binomial random graph. We introduce a general approach for…

We study sufficient conditions for the generic rigidity of a graph $G$ expressed in terms of (i) its minimum degree $\delta(G)$, or (ii) the parameter $\eta(G)=\min_{uv\notin E}(\deg(u)+\deg(v))$. For each case, we seek the smallest…

Let $G$ be a $k$-degenerate graph of order $n.$ It is well-known that $G\ $has no more edges than $S_{n,k},$ the join of a complete graph of order $k$ and an independent set of order $n-k.$ In this note it is shown that $S_{n,k}$ is…

A proper total $k$-colouring of a graph $G=(V,E)$ is an assignment $c : V \cup E\to \{1,2,\ldots,k\}$ of colours to the edges and the vertices of $G$ such that no two adjacent edges or vertices and no edge and its end-vertices are…

Let $M_{G}$ be the centered Hardy-Littlewood maximal operator on a finite graph $G$. We find $\underset{p\to \infty}{\lim}\|M_{G}\|_{p}^{p }$ when $G$ is the start graph ($S_n$) and the complete graph ($K_n$), and we fully describe…

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…

We consider the problem of realizable interval-sequences. An interval sequence comprises of $n$ integer intervals $[a_i,b_i]$ such that $0\leq a_i \leq b_i \leq n-1$, and is said to be graphic/realizable if there exists a graph with degree…

Let $G$ be an infinite graph whose vertex set is the set of positive integers, and let $G_n$ be the subgraph of $G$ induced by the vertices $\{1,2, \dots , n \}$. An increasing path of length $k$ in $G$, denoted $I_k$, is a sequence of…

We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$…

For a graph $F$, the $k$-subdivision of $F$, denoted $F^k$, is the graph obtained by replacing the edges of $F$ with internally vertex-disjoint paths of length $k$. In this paper, we prove that…

In this paper, we prove that every graph with average degree at least $s+t+2$ has a vertex partition into two parts, such that one part has average degree at least $s$, and the other part has average degree at least $t$. This solves a…

For given a graph $H$, a graphic sequence $\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\pi$ containing $H$ as a subgraph. In this paper, we characterize the potentially $H$-graphic…

The ($p$,1)-total number $\lambda_p^T(G)$ of a graph $G$ is the width of the smallest range of integers that suffices to label the vertices and the edges of $G$ such that no two adjacent vertices have the same label, no two incident edges…