We determine the maximum number of edges of an n-vertex graph G with the property that none of its r-cliques intersects a fixed set M ⊆ V (G). For (r−1)|M | > n, the (r − 1)-partite Turán graph turns out to be the unique extremal graph. For (r − 1)|M | < n, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results. Supported...

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is de6ned by ig(D) = max{d+(x); d−(x)} − min{d+(y); d−(y)} over all vertices x and y o...

Let $C_k$ be a cycle of order $k$, where $k\ge 3$. ex$(n, n, \{C_{3}, C_{4}\})$ the maximum number edges in balanced $3$-partite graph whose vertex set consists three parts, each has $n$ vertices that no subgraph isomorphic to $C_3$ or $C_4$. We construct dense 3-partite graphs without 3-cycles 4-cycles and show C_{4}\})\ge (\frac{6\sqrt{2}-8}{(\sqrt{2}-1)^{3/2}}+o(1))n^{3/2}$.

In this paper we study the length of the longest induced cycle in the unit circulant graph Xn = Cay(Zn; Z∗n), where Z∗n is the group of units in Zn. Using residues modulo the primes dividing n, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of ...

We investigate certain structural properties of random interdependent networks. We start by studying a property known as r-robustness, which is a strong indicator of the ability of a network to tolerate structural perturbations and dynamical attacks. We show that random k-partite graphs exhibit a threshold for r-robustness, and that this threshold is the same as the one for the graph to have mi...

The case k = 3 of a complete k-partite graph is called tripartite Tp,q,r. It that its vertices are decomposed into three disjoint sets such that, no two within the same set adjacent. has recently attracted much attention due to important in several applications, especially, chemistry where some molecular orbital compounds correspondents structure. One method capturing structure through computin...

In this paper we study the length of the longest induced cycle in the unitary Cayley graph Xn = Cay(Zn;Un), where Un is the group of units in Zn. Using residues modulo the primes dividing n, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of eac...

Constructions of woven graph codes based on constituent convolutional codes are studied, and examples of woven convolutional graph codes are presented. Existence of codes satisfying the Costello lower bound on the free distance within a random ensemble of woven graph codes based on s-partite, s-uniform hypergraphs is shown, where s depends only on the code rate. Simulation results for Viterbi d...

The chromatic capacity χcap(G) of a graph G is the largest k for which there exists a k-coloring of the edges of G such that, for every coloring of the vertices of G with the same colors, some edge is colored the same as both its vertices. We prove that there is an unbounded function f :N → N such that χcap(G) ≥ f(χ(G)) for almost every graph G, where χ denotes the chromatic number. We show tha...