Placing Two Edge-disjoint Copies of a Tree into a Bipartite Graph

•We study the existence of two packings of a tree into a bipartite graph with restrained maximum degree.•We present a constructive proof and our proof is Algorithmic.•Our proof maybe applied for the study of two packings of a tree into a balanced bipartite graph. We state that τ is an embedding of bipartite graph G(X0,X1) in the complete bipartite graph Bn(Y0,Y1) provided τ:V(G)→V(Bn), with σ(Xi)⊆Yi(i=0,1). Suppose that there are two embeddings of G in Bn such that both imagines under these two embeddings are edge-disjoint, we called that there is 2-packing of G in Bn. Let G(X1,X2) be a bipartite graph. For i=1,2, we use Δi to denote the maximum degree of the vertex in Xi. Let T(V1,V2) be a tree of order n with |V1|=a and |V2|=b. We demonstrate that if b≥a−1, there exists a 2-packing (σ,τ) of T in some Bn+1 such that Δ2(σ(T)∪τ(T))≤Δ2(T)+2. In general, Δ2(T)+2 can not be reduced to Δ2(T)+1, making this result sharp.