A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is ‘pseudorandom’, meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.
For a graph G and a non-negative integral weight function w on the vertex set of G, a set S of vertices of G is w-safe if w(C) ≥ w(D) for every component C of the subgraph of G induced by S and every component D of the subgraph of G induced by the complement of S such that some vertex in C is adjacent to some vertex of D. The minimum weight w(S) of a w-safe set S is the safe number s(G, w) of the weighted graph (G, w), and the minimum weight of a w-safe set that induces a connected subgraph of G is its connected safe number cs(G, w). Bapat et al. showed that computing cs(G, w) is NP-hard even when G is a star. For a given weighted tree (T, w), they described an efficient 2-approximation algorithm for cs(T, w) as well as an efficient 4-approximation algorithm for s(T, w). Addressing a problem they posed, we present a PTAS for the connected safe number of a weighted tree. Our PTAS partly relies on an exact pseudopolynomial time algorithm, which also allows to derive an asymptotic FPTAS for restricted instances. Finally, we extend a bound due to Fujita et al. from trees to block graphs.
For a graph G and an integer-valued threshold function τ on its vertex set, a dynamic monopoly is a set of vertices of G such that iteratively adding to it vertices u of G that have at least τ (u) neighbors in it eventually yields the vertex set of G. We show that the problem of finding a dynamic monopoly of minimum order can be solved in polynomial time for interval graphs with bounded threshold functions, but is NP-hard for chordal graphs allowing unbounded threshold functions.
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills a boundedness condition that is asymptotically best possible.
This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and graph labellings.
For a graph G and p ∈ [0, 1], let Gp arise from G by deleting every edge mutually independently with probability 1 − p. The random graph model (Kn)p is certainly the most investigated random graph model and also known as the G(n, p)-model. We show that several results concerning the length of the longest path/cycle naturally translate to Gp if G is an arbitrary graph of minimum degree at least n − 1.For a constant c, we show that asymptotically almost surely the length of the longest path is at least (1−(1+ǫ(c))ce −c )n for some function ǫ(c) → 0 as c → ∞, and the length of the longest cycle is a least (1 − O(c − 1 5 ))n. The first result is asymptotically best-possible. This extents several known results on the length of the longest path/cycle of a random graph in the G(n, p)-model.
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