The complete multipartite graph Kn(2t) having n partite sets of size 2t, with n ≥ 6 and t ≥ 1, is shown to have a decomposition into gregarious 6-cycles, that is, the cycles which have at most one vertex from any particular partite set. Complete sets of differences of numbers in Zn are used to produce starter cycles and obtain other cycles by rotating the cycles around the n-gon of the partite ...

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of hypergraphs $\mathcal{F}$, the Turán number ex$_r^{lin}(n,\mathcal{F})$ maximum on $n$ vertices that does not contain any member $\mathcal{F}$ as subgraph.
 Let $K_l$ be complete graph with $l$ and $r\geq 2$. The $r$-expansion $r$-graph $K_l^+$ obtained from by enlarging each edge $r-2...

Let G,J n) be an r-chromatic graph with n vertices in each colour class . Suppose G = G 3 (n), and t (G) . the minimal degree in G, is at least n + t (t _> 1) . We prove that C contains at least t 3 triangles but does not have to contain more titan 4t 3 of them . Furthermore, we give lower bounds for s such that G contains a complete 3-partite graph with s vertices in each class . Let';.(ii) = ...

A famous conjecture (usually called Ryser’s conjecture) that appeared in the PhD thesis of his student, J. R. Henderson, states that for an r-uniform r-partite hypergraph H, the inequality τ(H) 6 (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, and in the case of r = 3, which was proved by Aharoni in 2001. Here we ...

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...

A cycle in a multipartite graph G is gregarious if it contains at most one vertex from each partite set of G. The complete n-partite graph with partite sets of size m, denoted by Kn(m) is shown to have a decomposition into gregarious 4-cycles. The notion of a gregarious 4-cycle decomposition of this type was introduced in [3]. A 4-cycle decomposition of Kn(m) is circular if it is it is invarian...

It is known that for a graph on n vertices bn2/4c + 1 edges is sufficient for the existence of many triangles. In this paper, we determine the minimum number of edges sufficient for the existence of k triangles intersecting in exactly one common vertex. 1 Notation With integers n ≥ p ≥ 1, we let Tn,p denote the Turán graph, i.e., the complete p-partite graph on n vertices where each partite set...

We study a game on a graph G played by r revolutionaries and s spies. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move, and then each spy has the same option. The revolutionaries win if m of them meet at some vertex having no spy (at the end of a round); the spies win if they can avoid this forev...

For an ordered subset W={w1,w2,…,wk} of vertices and a vertex v in connected graph G, the k-vector r(v|W)=(d(v,w1),d(v,w2),…,d(v,wk)) is called representation with respect to W, where d(v,wi) distance between wi, for 1≤i≤k. The set W resolving G if r(u|W)≠r(v|W), every pair u,v∈V(G). minimum positive integer k which has cardinality metric dimension denoted as dim(G). A dim(G) basis G. bipartite...