The classical stability theorem of Erdős and Simonovits states that, for any fixed graph with χ(H) = k+ 1 ≥ 3, the following holds: every n-vertex graph that is H-free and has within o(n2) of the maximal possible number of edges can be made into the k-partite Turán graph by adding and deleting o(n2) edges. In this paper, we prove sharper quantitative results for graphs H with a critical edge, b...

We prove that if $G$ is a $k$-partite graph on $n$ vertices in which all of the parts have order at most $n/r$ and every vertex adjacent to least $1-1/r+o(1)$ proportion other part, then contains $(r-1)$-st power Hamiltonian cycle

An ordered r-graph is an r-uniform hypergraph whose vertex set linearly ordered. Given 2?k?r, H interval k-partite if there exist at least k disjoint intervals in the ordering such that every edge of has nonempty intersection with each and contained their union. Our main result implies ?>k?1, then for d>0 n-vertex dn? edges some m?n m-vertex subgraph ?(dm?) edges. This extension to r-graphs obs...

Let G be a 3-partite graph with 3n vertices, n in each class, such that each vertex is connected to at least 2 3 n+ 2 √ n of the vertices in each of the other two classes. We prove that G contains n independent triangles.

The star graph has been recognized as an attractive alternative to the hypercube. Let Fe and Fv be the sets of vertex faults and edge faults, respectively. Previously, Tseng et al. showed that an n-dimensional star graph can embed a ring of length n! if |Fe |≤n-3 (|Fv|=0), and a ring of length at least n!-4|Fv | if |Fv |≤n-3 (|Fe |=0). Since an n-dimensional star graph is regular of degree n-1 ...

Generally, a graph G, an independent set is a subset S of vertices in G such that no two vertices in S are adjacent (connected by an edge) and a vertex cover is a subset S of vertices such that each edge of G has at least one of its endpoints in S. Again, the minimum vertex cover problem is to find a vertex cover with the smallest number of vertices. Consider a k-partite graph G = (V, E) with v...

The proof of Theorem 1 consists of the following two lemmas. Recall that K r is the complete r-partite graph with p vertices in each class. In other words, K r = Tr(pr), the Turán graph with pr many vertices. It is easy to see that χ(K r ) = r. Lemma 2. For all c, η > 0, n > 8/η, if G is a graph on n vertices with e(G) ≥ (c + η) ( n 2 ) , then G has a subgraph G′ with n′ ≥ 1 2 √ ηn vertices suc...

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 defined by ig(D) = max{d+(x), d−(x)}−min{d+(y), d−(y)} over all vertices x and y of D (including x = y). If ig(D) = 0, then D is regular and if ig(D) ≤ 1, then D is almost regular. A c-partite tournament is an orientation of a complete c-...

By using the Szemerédi Regularity Lemma [10], Alon and Sudakov [1] recently extended the classical Andrásfai-Erdős-Sós theorem [2] to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r+1)-partite graph H whose smallest part has t vertices, there exists a constant C such that for any given ε > 0 and sufficiently large n...

The following sharpening of Turán’s theorem is proved. Let Tn,p denote the complete p– partite graph of order n having the maximum number of edges. If G is an n-vertex Kp+1-free graph with e(Tn,p) − t edges then there exists an (at most) p-chromatic subgraph H0 such that e(H0) ≥ e(G)− t. Using this result we present a concise, contemporary proof (i.e., one using Szemerédi’s regularity lemma) fo...