Packing k‐edge trees in graphs of restricted vertex degrees

Abstract

AbstractLet ${\cal G}^{s}_{r}$ denote the set of graphs with each vertex of degree at least r and at most s, v(G) the number of vertices, and τk (G) the maximum number of disjoint k‐edge trees in G. In this paper we show that if G ∈ ${\cal G}^{s}_{2}$ and s ≥ 4, then τ2(G) ≥ v(G)/(s + 1), if G ∈ ${\cal G}^{3}_{2}$ and G has no 5‐vertex components, then τ2(G) ≥ v(G)4, if G ∈ ${\cal G}^{s}_{1}$ and G has no k‐vertex component, where k ≥ 2 and s ≥ 3, then τk(G) ≥ (v(G) ‐k)/(sk ‐ k + 1), and the above bounds are attained for infinitely many connected graphs. Our proofs provide polynomial time algorithms for finding the corresponding packings in a graph. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 306–324, 2007