A graph G is called F -saturated if it is F -free but the addition of any missing edge to G creates a copy of F . Let the saturation function sat(n, F ) be the minimum number of edges that an F -saturated graph on n vertices can have. We determine this function asymptotically for every fixed complete partite graph F as n tends to infinity and give some structural information about almost extrem...

A bipartite graph is hamiltonian laceable if there exists a hamiltonian path between any two vertices that are in different partite sets. A hamiltonian laceable graph G is said to be hyperhamiltonian laceable if, for any vertex v of G, there exists a hamiltonian path of G−{v} joining any two vertices that are located in the same partite set different from that of v. In this paper, we further im...

For every real p > 0 and simple graph G, set f (p,G) = ∑ u∈V (G) d (u) , and let φ (r, p, n) be the maximum of f (p,G) taken over all Kr+1-free graphs G of order n. We prove that, if 0 < p < r, then φ (r, p, n) = f (p, Tr (n)) , where Tr (n) is the r-partite Turan graph of order n. For every p ≥ r + ⌈√ 2r ⌉ and n large, we show that φ (p, n, r) > (1 + ε) f (p, Tr (n)) for some ε = ε (r) > 0. Ou...

In this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: Let n ≥ 5 be an arbitrary positive odd integer; then, in any two-coloring of the edges of the complete 5-partite graph K((n − 1)/2, (n − 1)/2, (n − 1)/2, (n − 1)/2, 1) there is a monochromatic Cn, ∗2000 Mathematics Subject Classification: 05C55, 05C38. †Research supported in part by the Nat...

This paper provides lower orientable k-step domination number and upper orientable k-step domination number of complete r-partite graph for 1 ≤ k ≤ 2. It also proves that the intermediate value theorem holds for the complete r-partite graphs.

Let the k-graph Fan consist of k edges that pairwise intersect exactly in one vertex x, plus one more edge intersecting each of these edges in a vertex different from x. We prove that, for n sufficiently large, the maximum number of edges in an n-vertex k-graph containing no copy of Fan is ∏ k i=1 ⌊ k ⌋, which equals the number of edges in a complete k-partite k-graph with almost equal parts. T...

Let Kn,n denote the complete bipartite graph with n vertices in each partite set and Kn,n+I denote Kn,n with a one-factor added. It is proved in this paper that there exists an m-cycle system of Kn,n + I if and only if n ≡ 1 (mod 2), m ≡ 0 (mod 2), 4 ≤ m ≤ 2n, and n(n + 1) ≡ 0 (mod m).

Let r ≥ 2 and c > 0. Every graph on n vertices with at least cnr cliques on r vertices contains a complete r-partite subgraph with r − 1 parts of size ⌊cr log n⌋ and one part of size greater than n1−c r−1 . This result implies the Erdős-Stone-Bollobás theorem, the essential quantitative form of the Erdős-Stone theorem.

Abstract The book graph $B_n ^{(k)}$ consists of $n$ copies $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ Here consider natural off-diagonal variant $r(B_{cn} B_n^{(k)})$ for fixed $c \in (0,1]$ more general setting, show that an interesting dichotomy emerges: very small $c$ , simple $k$ -partite construction ...

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, ...