A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphsKp1,p2,...,pr =Ka1·p1,a2·p2,...,as ·ps with s=3, 4.We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s = 4, we giv...

For n > r > 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete r-graph on n vertices into q complete r-partite r-graphs. Graham and Pollak showed that fz(n) = n-1. Here we observe that f3(n) = n-2 and show that for every fixed r > 2, there are positive constants cx(r) and c2(r) such that q(r) < f,(n)" n-f'/2J < c2(r) for all n > r. This sol...

Given a graph property P, it is interesting to determine the typical structure of graphs that satisfy P. In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs. Using results from the theory of graph limits, we show that if P is a monotone property and r is the largest integer for which every r-colorable graph satis es P, then almost every gra...

For an r-uniform hypergraph H, let f(H) be the minimum number of complete r-partite r-uniform subhypergraphs of H whose edge sets partition the edge set of H. In this talk, I will discuss the value of f(H) for the random hypergraph H. Namely, I will prove that if (log n)/n ≤ p ≤ 1/2 and H ∈ H(n, p), then with high probability f(H) = (1− π(K(r−1) r ) + o(1)) ( n r−1 ) , where π(K(r−1) r ) is the...

In this paper we prove an asymptotic multipartite version of a well-known theorem of Kühn and Osthus by establishing, for any graph H with chromatic number r, the asymptotic multipartite minimum degree threshold which ensures that a large r-partite graph G admits a perfect H-tiling. We also give the threshold for an H-tiling covering all but a linear number of vertices of G, in a multipartite a...

In this paper, we characterize the shellable complete $t$-partite graphs. We also show for these types of graphs the concepts vertex decomposable, shellable and sequentially Cohen-Macaulay are equivalent. Furthermore, we give a combinatorial condition for the Cohen-Macaulay complete $t$-partite graphs.

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...

Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of H-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph H. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical H-...

A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of G are colored the same. For a κ-connected graph G and an integer k with 1 ≤ k ≤ κ, the rainbow k-connectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edgecoloring of G such that every two distinct vertices of G are connected by k interna...