A multiple-interval representation of a graph G is a mapping f which assigns to each vertex of G a union of intervals on the real line so that two distinct vertices u and v are adjacent if and only if f(u) ∩ f(v) = ∅. We study the total interval number of G, de0ned as I(G) = min {∑ v∈V #f(v): f is a multiple-interval representation of G } ; where #f(v) is the minimum number of intervals whose u...

In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r ∈ N, almost every n-vertex Kr+1-free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r 6 (log n). The proof combines a new (close to sharp) supersaturation version of the Erdős–Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, ...

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-...

Erdős, Rubin, and Taylor found a nice correspondence between the minimum order of a complete bipartite graph that is not r-choosable and the minimum number of edges in an r-uniform hypergraph that is not 2-colorable (in the ordinary sense). In this note we use their ideas to derive similar correspondences for complete kpartite graphs and complete k-uniform k-partite hypergraphs.

A digraph obtained by replacing each edge of a complete p-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicom-plete p-partite digraph, or just a semicomplete multipartite digraph. A semicomplete multipartite digraph with no cycle of length two is a multipartite tournament. In a digraph D, an r-king is a vertex q such that every vertex in D ...

In this paper, we tighten the concise Turán theorem for irregular graphs, using spectral and non-spectral proofs. We then investigate to what extent Turán’s theorem can be similarly strengthened for generalized r-partite graphs.

We introduce a search problem for finding a regular bi-partite graph of maximum attainable girth for specified degree and number of vertices, by restricting the search space using a series of mathematically rigourous arguments from [1] and [2]. The goal of this paper is to derive the enumeration search algorithm for finding a girth maximum (m, r) BTU, which is notation for regular partite graph...

Many studies are aimed at identifying dense clusters/subgraphs from protein-protein interaction (PPI) networks for protein function prediction. However, the prediction performance based on the dense clusters is actually worse than a simple guilt-by-association method using neighbor counting ideas. This indicates that the local topological structures and properties of PPI networks are still open...

A famous conjecture (usually called Ryser’s conjecture), appeared in the Ph.D thesis of his student, J. R. Henderson [9], states that for an r-uniform r-partite hypergraph H, the inequality τ(H) ≤ (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem [16], and in the case of r = 3, which was proved by Aharoni in 2001 [2]...

An n-partite tournament is an orientation of a complete n-partite graph. An npartite tournament is a tournament, if it contains exactly one vertex in each partite set. Douglas, Proc. London Math. Soc. 21 (1970) 716-730, obtained a characterization of strongly connected tournaments with exactly one Hamilton cycle (i.e., n-cycle). For n ≥, we characterize strongly connected n-partite tournaments ...