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 ∈ VG the sum of values x(e) for all edges e incident to v does not exceed 2. The size of x is the sum ∑e x(e). If {e ∈ 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 maximum triangle free 2-matching of 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 these 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 [Formula: see text] 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 + n3/2)-algorithm for d = 2k(k ≥ 2), and an O(n2)-algorithm for d = 2k + 1(k ≥ 2). We also prove that there exists a constant c > 1 such that every 3-regular graph contains at least cn perfect triangle-free 2-matchings.