Koh and Tan gave a sufficient condition for a 3-partite tournament to have at least one 3-king in [K.M. Koh, B.P. Tan, Kings in multipartite tournaments, Discrete Math. 147 (1995) 171–183, Theorem 2]. In Theorem 1 of this paper, we extend this result to n-partite tournaments, where n 3. In [K.M. Koh, B.P. Tan, Number of 4-kings in bipartite tournaments with no 3-kings, Discrete Math. 154 (1996)...

A sequence S is potentiallyKp1,p2,...,pt graphical if it has a realization containing aKp1,p2,...,pt as a subgraph, whereKp1,p2,...,pt is a complete t-partite graph with partition sizes p1, p2, ..., pt(p1 ≥ p2 ≥ ... ≥ pt ≥ 1). Let σ(Kp1,p2,...,pt, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Kp1,p2,...,pt, n) is potentially Kp1,p2,...,pt graphical....

Let T be a 3-partite tournament. We say that a vertex v is −→ C3-free if v does not lie on any directed triangle of T . Let F3(T ) be the set of the −→ C3-free vertices in a 3-partite tournament and f3(T ) its cardinality. In a recent paper, it was proved that if T is a regular 3-partite tournament, then f3(T ) < n 9 , where n is the order of T . In this paper, we prove that f3(T ) ≤ n 12 . We ...

Let T be a 3-partite tournament. We say that a vertex v is −→ C3 -free if v does not lie on any directed triangle of T . Let F3(T ) be the set of the −→ C3 -free vertices in a 3-partite tournament and f3(T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F3(T )must be contained in one of the partite sets of T . It is also shown that for every regular 3...

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...

Let F be a graph, k ≥ 2 be an integer, and write exχ≤k(n, F ) for the maximum number of edges in an n-vertex graph that is k-partite and has no subgraph isomorphic to F . The function exχ≤2(n, F ) has been studied by many researchers. Finding exχ≤2(n,Ks,t) is a special case of the Zarankiewicz problem. We prove an analogue of the Kövári-Sós-Turán Theorem by showing exχ≤3(n,Ks,t) ≤ ( 1 3 )1−1/s(...

The Erdős–Hajnal Theorem asserts that non-universal graphs, is, graphs do not contain an induced copy of some fixed graph H, have homogeneous sets size significantly larger than one can generally expect to find in a graph. We obtain two results this flavor the setting r-uniform hypergraphs. A theorem Rödl if n-vertex is then it contains almost set (i.e. with edge density either very close 0 or ...

A multipartite or c-partite tournament is an orientation of a complete c-partite graph. Lu and Guo (submitted for publication) [3] recently introduced strong quasi-Hamiltonianconnectivity of a multipartite tournament D as follows: For any two distinct vertices x and y ofD, there is a pathwith at least one vertex from each partite set ofD from x to y and from y to x. We obtain the definition for...

A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a nite number of isolated vertices to it will always yield a sum graph, and the sum number (G) of G is the smallest number of isolated vertices that...

Let G be a k-partite graph with n vertices in parts such that each vertex is adjacent to at least δ∗(G) vertices in each of the other parts. Magyar and Martin [20] proved that for k = 3, if δ∗(G) ≥ 2 3 n and n is sufficiently large, then G contains a K3-factor (a spanning subgraph consisting of n vertex-disjoint copies of K3) except that G is one particular graph. Martin and Szemerédi [21] prov...