In [8], it is shown that the complete multipartite graph Kn(2t) having n partite sets of size 2t, where n ≥ 6 and t ≥ 1, has a decomposition into gregarious 6-cycles if n ≡ 0, 1, 3 or 4 (mod 6). Here, a cycle is called gregarious if it has at most one vertex from any particular partite set. In this paper, when n ≡ 0 or 3 (mod 6), another method using difference set is presented. Furthermore, wh...

A graph G is said to be F-saturated if G does not contain a copy of F as a subgraph and G+ e contains a copy of F as a subgraph for any edge e contained in the complement of G. Erdős et al. in [A problem in graph theory, Amer. Math. Monthly 71 (1964) 1107–1110.] determined the minimum number of edges, sat(n, F ), such that a graph G on n vertices must have when F is a t-clique. Later, Ollmann [...

Let $C_k$ be a cycle of order $k$, where $k\ge 3$. ex$(n, n, \{C_{3}, C_{4}\})$ the maximum number edges in balanced $3$-partite graph whose vertex set consists three parts, each has $n$ vertices that no subgraph isomorphic to $C_3$ or $C_4$. We construct dense 3-partite graphs without 3-cycles 4-cycles and show C_{4}\})\ge (\frac{6\sqrt{2}-8}{(\sqrt{2}-1)^{3/2}}+o(1))n^{3/2}$.

An orientation of a complete graph is a tournament, and an orientation of a complete n-partite graph is an n-partite tournament. For each n 2:: 4, there exist examples of strongly connected n-partite tournament without any strongly connected subtournaments of order p 2:: 4. If D is a digraph, then let d+ (x) be the out degree and d(x) the indegree of the vertex x in D. The minimum (maximum) out...

We show that every K4-free graph G with n vertices can be made bipartite by deleting at most n/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős.

The cyclicity of a graph is the largest integer n for which the graph is contractible to the cycle on n vertices. By analyzing the cycle space of a graph, we establish upper and lower bounds on cyclicity. These bounds facilitate the computation of cyclicity for several classes of graphs, including chordal graphs, complete n-partite graphs, n-cubes, products of trees and cycles, and planar graph...

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...

We prove that, for r ≥ 2 and n ≥ n(r), every directed graph with n vertices and more edges than the r-partite Turán graph T (r, n) contains a subdivision of the transitive tournament on r + 1 vertices. Furthermore, the extremal graphs are the orientations of T (r, n) induced by orderings of the vertex classes.

We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 in which this upper bound is attained, this graph can be viewed as a generalization of the ...