Let f(n, r, k) be the minimal number such that every hypergraph larger than f(n, r, k) contained in ([n] r ) contains a matching of size k, and let g(n, r, k) be the minimal number such that every hypergraph larger than g(n, r, k) contained in the r-partite r-graph [n]r contains a matching of size k. The Erdős-Ko-Rado theorem states that f(n, r, 2) = (n−1 r−1 ) (r ≤ n 2 ) and it is easy to show...

In this paper, for any prime p 11, we consider Cp-decompositions of Km × Kn and Km ∗ Kn and also Cp-factorizations of Km × Kn, where × and ∗ denote the tensor product and wreath product of graphs, respectively, (Km ∗ Kn is isomorphic to the complete m-partite graph in which each partite set has exactly n vertices). It has been proved that for m, n 3, Cp-decomposes Km ×Kn if and only if (1) eith...

Turán’s Theorem states that every graphG of edge density ‖G‖/ (|G| 2 ) > k−2 k−1 contains a complete graph K and describes the unique extremal graphs. We give a similar Theorem for `-partite graphs. For large `, we find the minimal edge density d` , such that every `-partite graph whose parts have pairwise edge density greater than d` contains a K . It turns out that d` = k−2 k−1 for large enou...

Let K(X; Y ) denote the bipartite framework in the plane that realizes the complete bipartite graph Km;n with partite sets X; Y ; |X | = m; |Y | = n. We show that for m¿3; n¿5, K(X; Y ) admits a continuous deformation if and only if X lies on a line ‘ and Y lies on a line perpendicular to ‘. c © 2001 Elsevier Science B.V. All rights reserved.

The Andrásfai–Erdős–Sós Theorem [2] states that all triangle-free graphs on n vertices with minimum degree strictly greater than 2n/5 are bipartite. Thomassen [11] proved that when the minimum degree condition is relaxed to ( 3 + ε)n, the result is still guaranteed to be rε-partite, where rε does not depend on n. We prove best possible random graph analogues of these theorems.

graph is called a complete r-partite graph if its vertex set can be partitioned into r independent sets V1, . . . , Vr such that for any i = 1, 2, . . . , r every vertex in Vi is adjacent to all other vertices in Vj , j 6= i. We denote a complete r-partite graph with part sizes |Vi| = ni by Kn1,...,nr . For a graph G and a positive integer k we use kG to denote k vertex-disjoint copies of G. Gi...

We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(G) to G, then the graph G+ e contains Kp. We study the minimum and maximum size of non-(p − 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: Unauthenticated Download Date | 2/2/17 7:37 PM 10 K. Amin, J. Faudree, R.J. Gould and E. Sidorowicz for which va...

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number ω, then the largest eigenvalue q (G) of the matrix Q = A+D satisfies q (G) ≤ 2 (1− 1/ω)n. If G is a complete regu...