Rolling backwards can move you forward: On embedding problems in sparse expanders

We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. This proves to be a powerful tool for embedding graphs of large girth into expander graphs. As an application of this method, we settle two problems: • For a graph H, we denote by H^q the graph obtained from H by subdividing its edges with q−1 vertices each. We show that the k-size-Ramsey number R^_k(H^q) satisfies R^_k(H^q) = O(qn) for every bounded degree graph H on n vertices and for q = Ω(log n), which is optimal up to a constant factor. This settles a conjecture of Pak (2002). • We give a deterministic, polynomial time algorithm for finding vertex-disjoint paths between given pairs of vertices in a strong expander graph. More precisely, let G be an (n, d, λ)-graph with λ = O(d¹−^ε), and let P be any collection of at most cⁿ_log^log_n^d disjoint pairs of vertices in G for some small constant c, such that in the neighborhood of every vertex in G there are at most d/4 vertices from P. Then there exists a polynomial time algorithm which finds vertex-disjoint paths between every pair in P, and each path is of the same length l = O ( ^log_logⁿ_d ). Both the number of pairs and the length of the paths are optimal up to a constant factor; the result answers the offline version of a question of Alon and Capalbo (2007).