Local algorithms, regular graphs of large girth, and random regular graphs

Abstract

We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This is an uncommon direction of transfer of results, which is usually from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As examples, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum and minimum bisection, maximum $k$-independent sets, minimum $k$-dominating sets and minimum connected and weakly-connected dominating sets in $r$-regular graphs with large girth.Comment: Third version: no changes were made to the file. We would like to point out that this paper was split into two parts in the publication process. General theorems are in a paper with the same title, accepted by Combinatorica. The applications of Section 9 are in a paper entitled "Properties of regular graphs with large girth via local algorithms", published by JCTB, doi 10.1016/j.jctb.2016.07.00