Vertex-Disjoint Paths in Graphs

Let n₁, n₂, . . ., n_k be integers of at least two. Johansson gave a minimum degree condition for a graph of order exactly n₁ + n₂ + ⋯ + n_k to contain k vertex-disjoint paths of order n₁, n₂, . . ., n_k, respectively. In this paper, we extend Johansson's result to a corresponding packing problem as follows. Let G be a connected graph of order at least n₁ + n₂ + ⋯ + n_k. Under this notation, we show that if the minimum degree sum of three independent vertices in G is at least (formula presented) then G contains k vertex-disjoint paths of order n₁, n₂, . . ., n_k, respectively, or else n₁ = n₂ = ⋯ = n_k = 3, or k = 2 and n₁ = n₂ = odd. The graphs in the exceptional cases are completely characterized. In particular, these graphs have more than n₁ + n₂ + ⋯ + n_k vertices.