Hamiltonian cycles that pass through of a linear forest in bipartitos grafos balanced

Abstract

Let G = (A ∪ B, E) be a bipartite graph whith |A| = |B| = n ≥ 4. A graph is linear forest if every component is a path. Let S be a set of medges of G that induces a linear forest. We prove that if σ₁,₁(G) = min{d_G(u) + d_G(v) : u ∈ A, v ∈ B, uv ̸∈ E(G)} ≥ (n+1)+m,  then G contains (m + 1) hamiltonian cycles C_j such that |E(C_j ) ∩ S| = j, with j = 0, 1, . . . , m.