the inflation $g_{i}$ of a graph $g$ with $n(g)$ vertices and $m(g)$ edges is obtained from $g$ by replacing every vertex of degree $d$ of $g$ by a clique, which is isomorph to the complete graph $k_{d}$, and each edge $(x_{i},x_{j})$ of $g$ is replaced by an edge $(u,v)$ in such a way that $uin x_{i}$, $vin x_{j}$, and two different edges of $g$ are replaced by non-adjacent edges of $g_{i}$. t...

The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...

In this paper we disprove a conjecture of Lidický and Murphy about the number copies given graph in $K_r$-free give an alternative general conjecture. We also prove asymptotically tight bound on any bipartite radius at most~$2$ triangle-free graph.

The feasible region $\Omega_{{\rm ind}}(F)$ of a graph $F$ is the collection points $(x,y)$ in unit square such that there exists sequence graphs whose edge densities approach $x$ and induced $F$-densities $y$. A complete description not known for any with at least four vertices clique or an independent set. provides lot combinatorial information about $F$. For example, supremum $y$ over all $(...

For a graph $G$, let $f(G)$ denote the size of maximum cut in $G$. The problem estimating as function number vertices and edges $G$ has long history was extensively studied last fifty years. In this paper we propose an approach, based on semidefinite programming, to prove lower bounds $f(G)$. We use approach find large cuts graphs with few triangles $K_r$-free graphs.

it is shown that when a special vertex stretching is applied to a graph, the cochordal cover number of the graph increases exactly by one, as it happens to its induced matching number and (castelnuovo-mumford) regularity. as a consequence, it is shown that the induced matching number and cochordal cover number of a special vertex stretching of a graph g are equal provided g is well-covered bipa...

It is shown that when a special vertex stretching is applied to a graph, the cochordal cover number of the graph increases exactly by one, as it happens to its induced matching number and (Castelnuovo-Mumford) regularity. As a consequence, it is shown that the induced matching number and cochordal cover number of a special vertex stretching of a graph G are equal provided G is well-covered bipa...

Abstract Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$ , where $1\le k_1<k_2<\cdots <k_r=k$ . Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$ we show that there is a paucity nondiagonal solutions Diophantine system _j({\mathbf x...

A graph G is fractional k-covered if for each edge e of G, there exists a fractional k-factor h, such that h(e) = 1. If k = 2, then a fractional k-covered graph is called a fractional 2-covered graph. The binding number bind(G) is defined as follows, bind(G) = min{ |NG(X)| |X| : Ø = X ⊆ V (G), NG(X) = V (G)}. In this paper, it is proved that G is fractional 2-covered if δ(G) ≥ 4 and bind(G) > 5...