A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs

Let G be a balanced bipartite graph of order 2n≥4, and let σ1,1(G) be the minimum degree sum of two non-adjacent vertices in different partite sets of G. In 1963, Moon and Moser proved that if σ1,1(G)≥n+1, then G is hamiltonian. In this note, we show that if k is a positive integer, then the Moon-Moser condition also implies the existence of a 2-factor with exactly k cycles for sufficiently large graphs. In order to prove this, we also give a σ1,1 condition for the existence of k vertex-disjoint alternating cycles with respect to a chosen perfect matching in G.