We prove that, for r ≥ 2 and n ≥ n(r), every directed graph with n vertices and more edges than the r-partite Turán graph T (r, n) contains a subdivision of the transitive tournament on r + 1 vertices. Furthermore, the extremal graphs are the orientations of T (r, n) induced by orderings of the vertex classes.

An r-cut of the complete r-uniform hypergraph Kr n is obtained by partitioning its vertex set into r parts and taking all edges that meet every part in exactly one vertex. In other words it is the edge set of a spanning complete r-partite subhypergraph of Kr n. An r-cut cover is a collection of r-cuts so that each edge of K r n is in at least one of the cuts. While in the graph case r = 2 any 2...

Let r ≥ 2 and c > 0. If G is a graph of order n and the largest eigenvalue of its adjacency matrix satisfies μ (G) ≥ (1− 1/r + c)n, then G contains a complete r-partite subgraph with r − 1 parts of size ⌊

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

We prove that for all r ≥ 2 and c > 0, every G graph of order n with at least cnr cliques of order r contains a complete r-partite graph with each part of size ⌊cr log n⌋ . This result implies a concise form of the Erdős-Stone theorem.

in this paper we study the coprime graph of a group $g$. the coprime graph of a group $g$, is a graph whose vertices are elements of $g$ and two distinct vertices $x$ and $y$ are adjacent iff $(|x|,|y|)=1$. in this paper we classify all the groups which the coprime graph is a complete r-partite graph or a planar graph. also we study the automorphism group of the coprime graph.

Motivated by an old problem known as Ryser’s Conjecture, we prove that for r = 4 and r = 5, there exists > 0 such that every r-partite r-uniform hypergraph H has a cover of size at most (r − )ν(H), where ν(H) denotes the size of a largest matching in H.

Let r ≥ 2 and c > 0. Every graph on n vertices with at least cnr cliques on r vertices contains a complete r-partite subgraph with r − 1 parts of size ⌊cr log n⌋ and one part of size greater than n1−c r−1 . This result implies the Erdős-Stone-Bollobás theorem, the essential quantitative form of the Erdős-Stone theorem.

A latin transversal in a square matrix of order n is a set of entries, no two in the same row or column, which are pairwise distinct. A longstanding conjecture of Ryser states that every Latin square with odd order has a latin transversal. Some results on the existence of a large partial latin transversal can be found in [11,6,16]. Mainly motivated by Ryser’s conjecture, Erdős and Spencer [8] p...