Graph saturation in multipartite graphs

Abstract

Let G be a fixed graph and let F be a family of graphs. A subgraph J of G is F-saturated if no member of F is a subgraph of J, but for any edge e in E(G) − E(J), some element of F is a subgraph of J + e. We let ex(F, G) and sat(F, G) denote the maximum and minimum size of an F-saturated subgraph of G, respectively. If no element of F is a subgraph of G, then sat(In this paper, for k ≥ 3 and n ≥ 100 we determine sat(K 3 , K n k ), where K n k is the complete balanced k-partite graph with partite sets of size n. We also give several families of constructions of K t -saturated subgraphs of K n k for t ≥ 4. Our results and constructions provide an informative contrast to recent results on the edge-density

Partite Saturation of Complete Graphs

“…The bounds in (i), together with Theorem 1, imply that sat(n, k, r) = O(krn), answering a question of Ferrara, Jacobson, Pfender and Wenger [7]. In (ii), we determine exactly α(k, r) for some values of r and every k large enough, allowing us to disprove a conjecture in [7] which states that sat(n, k, r) = (k − 1)(2r − 3)n − (2r − 3)(r − 1) for k ≥ 2r − 3 and sufficiently large n. In (iii), we deal with the cases r = 3, 4, 5 which have not been covered by (ii). Finally, (iv) shows that the lower bound in (i), which is attained for certain values of r and k mentioned in (ii), is not tight when k = r.…”

“…In this paper, we are interested in the saturation number sat(n, k, r) = sat(K k×n , K r ) for k ≥ r ≥ 3 where K k×n is the complete k-partite graph containing n vertices in each of its k parts. This function was first studied recently by Ferrara, Jacobson, Pfender and Wenger [7] who determined sat(n, k, 3) for n ≥ 100. Later, Roberts [13] showed that sat(n, 4, 4) = 18n − 21 for sufficiently large n.…”

Partite Saturation Problems

“…In the bipartite case Bollobás [1,2] and Wessel [9,10] independently determined the saturation number sat(K a,b , K c,d ). Working in the r-partite setting with r 3, Ferrara, Jacobson, Pfender, and Wenger determined in [5] the value of sat(K 3 , K n r ) for sufficiently large n and showed that sat(K 3 , K n 3 ) = 6n − 6 for all n.…”

“…We consider the problem of determining the value sat p (H, Note that for a graph H with no homomorphism onto any proper subgraph of itself we have by definition sat p (H, H[n]) = sat(H, H[n]). In this way we know that sat p (K 3 , K 3 [n]) = 6n − 6 from [5] and can drop the partite requirement when considering cliques. Our main result, Theorem 1, is to show that for sufficiently large n we have sat(K 4 , K 4 [n]) = 18n−21.…”

“…We let H[n] denote the blow-up of H onto parts of size n and refer to a copy of H in H[n] as partite if it has one vertex in each part of H[n]. We then ask how few edges a subgraph G of H[n] can have such that G has no partite copy of H but such that the addition of any new edge from H[n] creates a partite H. When H is a triangle this value was determined by Ferrara, Jacobson, Pfender, and Wenger in [5]. Our main result is to calculate this value for H = K 4 when n is large.…”