Triangle-free 2-matchings revisited

Abstract

A 2-matching in an undirected graph G = (VG, EG) is a function x : EG → {0,1,2} such that for each node v ε VG the sum of values x(e) on all edges e incident to v does not exceed 2. The size of x is the sum σ e x(e). If {e ε EG | x(e) ≠ 0} contains no triangles then x is called triangle-free. Cornuéjols and Pulleyblank devised a combinatorial O(mn)-algorithm that finds a triangle free 2-matching of maximum size (hereinafter n :=|VG|, m :=|EG|) and also established a min-max theorem. We claim that this approach is, in fact, superfluous by demonstrating how their results may be obtained directly from the Edmonds-Gallai decomposition. Applying the algorithm of Micali and Vazirani we are able to find a maximum triangle-free 2-matching in -time. Also we give a short self-contained algorithmic proof of the min-max theorem. Next, we consider the case of regular graphs. It is well-known that every regular graph admits a perfect 2-matching. One can easily strengthen this result and prove that every d-regular graph (for d >3) contains a perfect triangle-free 2-matching. We give the following algorithms for finding a perfect triangle-free 2-matching in a d-regular graph: an O(n)-algorithm for d=3, an O(m+n 3/2)-algorithm for d=2k (k > 2), and an O(n 2)-algorithm for d=2k+1 (k >2). © 2010 Springer-Verlag Berlin Heidelberg.