An algorithmic Friedman-Pippenger theorem on tree embeddings and applications

An (n, d)-expander is a graph G = (V, E) such that for every X subset of V with vertical bar X vertical bar <= 2n - 2 we have vertical bar Gamma(G)(X) vertical bar >= (d + 1) vertical bar X vertical bar. A tree T is small if it has at most n vertices and has maximum degree at most d. Friedman and Pippenger (1987) proved that any ( n; d)- expander contains every small tree. However, their elegant proof does not seem to yield an efficient algorithm for obtaining the tree. In this paper, we give an alternative result that does admit a polynomial time algorithm for finding the immersion of any small tree in subgraphs G of (N, D, lambda)-graphs Lambda, as long as G contains a positive fraction of the edges of Lambda and lambda/D is small enough. In several applications of the Friedman-Pippenger theorem, including the ones in the original paper of those authors, the (n, d)-expander G is a subgraph of an (N, D, lambda)-graph as above. Therefore, our result suffices to provide efficient algorithms for such previously non-constructive applications. As an example, we discuss a recent result of Alon, Krivelevich, and Sudakov (2007) concerning embedding nearly spanning bounded degree trees, the proof of which makes use of the Friedman-Pippenger theorem. We shall also show a construction inspired on Wigderson-Zuckerman expander graphs for which any sufficiently dense subgraph contains all trees of sizes and maximum degrees achieving essentially optimal parameters. Our algorithmic approach is based on a reduction of the tree embedding problem to a certain on-line matching problem for bipartite graphs, solved by Aggarwal et al. (1996)

Expanders Are Universal for the Class of All Spanning Trees

Given a class of graphs F, we say that a graph G is universal for F, or F-universal, if every H in F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of attention. One is particularly interested in tight F-universal graphs, i.e., graphs whose number of vertices is equal to the largest number of vertices in a graph from F. Arguably, the most studied case is that when F is some class of trees. Given integers n and \Delta, we denote by T(n,\Delta) the class of all n-vertex trees with maximum degree at most \Delta. In this work, we show that every n-vertex graph satisfying certain natural expansion properties is T(n,\Delta)-universal or, in other words, contains every spanning tree of maximum degree at most \Delta. Our methods also apply to the case when \Delta is some function of n. The result has a few very interesting implications. Most importantly, we obtain that the random graph G(n,p) is asymptotically almost surely (a.a.s.) universal for the class of all bounded degree spanning (i.e., n-vertex) trees provided that p \geq c n^{-1/3} \log^2n where c > 0 is a constant. Moreover, a corresponding result holds for the random regular graph of degree pn. In fact, we show that if \Delta satisfies \log n \leq \Delta \leq n^{1/3}, then the random graph G(n,p) with p \geq c \Delta n^{-1/3} \log n and the random r-regular n-vertex graph with r \geq c\Delta n^{2/3} \log n are a.a.s. T(n,\Delta)-universal. Another interesting consequence is the existence of locally sparse n-vertex T(n,\Delta)-universal graphs. For constant \Delta, we show that one can (randomly) construct n-vertex T(n,\Delta)-universal graphs with clique number at most five. Finally, we show robustness of random graphs with respect to being universal for T(n,\Delta) in the context of the Maker-Breaker tree-universality game.Comment: 25 page

The approximate Loebl-Komlós-Sós conjecture I: The sparse decomposition

In a series of four papers we prove the following relaxation of the Loebl-Komlós-Sós conjecture: For every α > 0 there exists a number k0 such that for every k > k0, every n-vertex graph G with at least (1/2 + α)n vertices of degree at least (1 + α)k contains each tree T of order k as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemerédi regularity lemma: We decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. Since for sparse graphs G, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a suitable combinatorial structure inside the decomposition, which we then use for embedding the tree. © 2017 the authors

Size-Ramsey numbers of structurally sparse graphs

Size-Ramsey numbers are a central notion in combinatorics and have been widely studied since their introduction by Erd\H{o}s, Faudree, Rousseau and Schelp in 1978. Research has mainly focused on the size-Ramsey numbers of $n$-vertex graphs with constant maximum degree $\Delta$. For example, graphs which also have constant treewidth are known to have linear size-Ramsey numbers. On the other extreme, the canonical examples of graphs of unbounded treewidth are the grid graphs, for which the best known bound has only very recently been improved from $O(n^{3/2})$ to $O(n^{5/4})$ by Conlon, Nenadov and Truji\'c. In this paper, we prove a common generalization of these results by establishing new bounds on the size-Ramsey numbers in terms of treewidth (which may grow as a function of $n$). As a special case, this yields a bound of $\tilde{O}(n^{3/2 - 1/2\Delta})$ for proper minor-closed classes of graphs. In particular, this bound applies to planar graphs, addressing a question of Wood. Our proof combines methods from structural graph theory and classic Ramsey-theoretic embedding techniques, taking advantage of the product structure exhibited by graphs with bounded treewidth.Comment: 21 page