In this paper we propose a notion of pattern avoidance in binary trees that generalizes the avoidance of contiguous tree patterns studied by Rowland and non-contiguous tree patterns studied by Dairyko, Pudwell, Tyner, and Wynn. Specifically, we propose algorithms for generating different classes of binary trees that are characterized by avoiding one or more of these generalized patterns. This is achieved by applying the recent Hartung-Hoang-Mütze-Williams generation framework, by encoding binary trees via permutations. In particular, we establish a one-to-one correspondence between tree patterns and certain mesh permutation patterns. We also conduct a systematic investigation of all tree patterns on at most 5 vertices, and we establish bijections between pattern-avoiding binary trees and other combinatorial objects, in particular pattern-avoiding lattice paths and set partitions.

For integers ‍k≥ 1 and n≥ 2k+1, the Kneser graph ‍K(n,k) has as vertices all k-element subsets of an n-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph ‍K(5,2). This problem received considerable attention in the literature, including a recent solution for the sparsest case n=2k+1. The main contribution of this paper is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph ‍J(n,k,s) has as vertices all k-element subsets of an n-element ground set, and an edge between any two sets whose intersection has size exactly ‍s. Clearly, we have K(n,k)=J(n,k,0), i.e., generalized Johnson graph include Kneser graphs as a special case. Our results imply that all known families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász’ conjecture on Hamilton cycles in vertex-transitive graphs from ‍1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway’s Game of Life, and to analyze this system combinatorially and via linear algebra.