Some results on placing bipartite graphs of maximum degree two

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 u v ∈ E ). We prove that if G and H are two bipartite graphs of order | G | = | H | = 2 p ⩾ 4 such that the sizes of G and H satisfy ‖ G ‖ ⩽ 2 p − 3 and ‖ H ‖ ⩽ 2 p − 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. esult implies easily that for integers p and k 1 , k 2 , … , k l such that k i ⩾ 2 for i = 1 , … , l and k 1 + … + k 1 ⩽ p − 1 every bipartite balanced graph G of order 2p and size at least p 2 − 2 p + 3 contains mutually vertex disjoint cycles C 2 k 1 , … , C 2 k l , unless G = K 3 , 3 − 3 K 1 , 1 .