Finding any given 2‐factor in sparse pseudorandom graphs efficiently

Abstract

AbstractGiven an ‐vertex pseudorandom graph and an ‐vertex graph with maximum degree at most two, we wish to find a copy of in , that is, an embedding so that for all . Particular instances of this problem include finding a triangle‐factor and finding a Hamilton cycle in . Here, we provide a deterministic polynomial time algorithm that finds a given in any suitably pseudorandom graph . The pseudorandom graphs we consider are ‐bijumbled graphs of minimum degree which is a constant proportion of the average degree, that is, . A ‐bijumbled graph is characterised through the discrepancy property: for any two sets of vertices and . Our condition on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption‐reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work, incorporating the work of Nenadov, together with additional ideas and simplifications.