Independent cycles and paths in bipartite balanced graphs

Bipartite graphs G = (L, R; E) and H = (L, R; E) are bi-placeabe if there is a bijection f : L ∪ R → L ∪ R such that f(L) = L and f(u)f(v) / ∈ E for every edge uv ∈ E. We prove that if G and H are two bipartite balanced graphs of order |G| = |H | = 2p ≥ 4 such that the sizes of G and H satisfy ‖ G ‖≤ 2p− 3 and ‖ H ‖≤ 2p− 2, and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs. This result implies easily that for integers p and k1, k2, . . . , kl such that ki ≥ 2 for i = 1, . . . , l and k1 + · · · + kl ≤ p − 1 every bipartite balanced graph G of order 2p and size at least p − 2p + 3 contains mutually vertex disjoint cycles C2k1 , . . . , C2kl , unless G = K3,3−3K1,1.