Given a poset P, a family F of elements in the Boolean lattice is said to be P-saturated if (1) F contains no copy of P as a subposet and (2) every proper superset of F contains a copy of P as a subposet. The maximum size of a P-saturated family is denoted by La(n, P), which has been studied for a number of choices of P. The minimum size of a P-saturated family, sat(n, P), was introduced by Gerbner et al. (2013), and parallels the deep literature on the saturation function for graphs.We introduce and study the concept of saturation for induced subposets. As opposed to induced saturation in graphs, the above definition of saturation for posets extends naturally to the induced setting. We give several exact results and a number of bounds on the induced saturation number for several small posets. We also use a transformation to the biclique cover problem to prove a logarithmic lower bound for a rich infinite family of target posets. †

An r-augmented tree is a rooted tree plus r edges added from each leaf to ancestors. For d, g, r ∈ N, we construct a bipartite r-augmented complete d-ary tree having girth at least g. The height of such trees must grow extremely rapidly in terms of the girth.Using the resulting graphs, we construct sparse non-k-choosable bipartite graphs, showing that maximum average degree at most 2(k − 1) is a sharp sufficient condition for k-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-k-colorable graphs and hypergraphs with any girth g.

In this paper, we study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model.In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with m random vertices selected with probabilities proportional to their current degrees. (Constant m is the only parameter of the model.) We prove that if m ≥ 1253, then asymptotically almost surely there exists a perfect matching.

Given a family scriptF and a host graph H, a graph G⊆H is scriptF‐saturated relative to H if no subgraph of G lies in scriptF but adding any edge from E(H)−E(G) to G creates such a subgraph. In the scriptF‐saturation game on H, players Max and Min alternately add edges of H to G, avoiding subgraphs in scriptF, until G becomes scriptF‐saturated relative to H. They aim to maximize or minimize the length of the game, respectively; sat gfalse(scriptF;Hfalse) denotes the length under optimal play (when Max starts). Let scriptO denote the family of odd cycles and Tn the family of n‐vertex trees, and write F for scriptF when F={F}. Our results include sat gfalse(scriptO;Knfalse)=⌊n2⌋false⌈n2false⌉, sat gfalse(Tn;Knfalse)=false(0ptn−22false)+1 for n≥6, sat gfalse(K1,3;Knfalse)=2⌊n2⌋ for n≥8, and sat gfalse(P4;Knfalse)∈{⌊4n5⌋,⌈4n5⌉} for n≥5. We also determine sat gfalse(P4;Km,nfalse); with m≥n, it is n when n is even, m when n is odd and m is even, and m+⌊n/2⌋ when mn is odd. Finally, we prove the lower bound sat gfalse(C4;Kn,nfalse)≥121n13/12−Ofalse(n35/36false). The results are very similar when Min plays first, except for the P4‐saturation game on Km,n.

Abstract. We consider the effect on the length of the game of Cops and Robbers when more cops are added to the game play. In Overprescribed Cops and Robbers, as more cops are added, the capture time (the minimum length of the game assuming optimal play) monotonically decreases. We give the full range of capture times for any number of cops on trees, and classify the capture time for an asymptotic number of cops on grids, hypercubes, and binomial random graphs. The capture time of planar graphs with a number of cops at and far above the cop number is considered.