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 ${\displaystyle k_i\ge 2}$ for i = 1,...,l and k₁ +...+ kₗ ≤ p-1 every bipartite balanced graph G of order 2p and size at least p²-2p+3 contains mutually vertex disjoint cycles ${\displaystyle C_{2k₁},...,C_{2kₗ}}$, unless ${\displaystyle G=K_{3,3}-3K_{1,1}}$.