Search

In a seminal paper on finding large matchings in sparse random graphs, Karp and Sipser proposed two algorithms for this task. The second algorithm has been intensely studied, but due to technical difficulties, the first algorithm has received less attention. Empirical results by Karp and Sipser suggest that the first algorithm is superior. In this paper we show that this is indeed the case, at least for random k‐regular graphs. We show that w.h.p. the first algorithm will find a matching of size n/2−O(logn) in a random k‐regular graph, k = O(1). We also show that the algorithm can be adapted to find a maximum matching in O(n) time w.h.p. This is to be compared with O(n3/2) time for the worst‐case. [ABSTRACT FROM AUTHOR]

In this study we consider a Ramsey property of random d‐regular graphs, G(n,d). Let r≥2 be fixed. Then w.h.p. the edges of G(n,2r) can be colored such that every monochromatic component has order o(n). On the other hand, there exists a constant γ>0 such that w.h.p., every r‐coloring of the edges of G(n,2r+1) must contain a monochromatic cycle of length at least γn. We prove an analogous result for random k‐out graphs. [ABSTRACT FROM AUTHOR]