Degree sequence realizations with given packing and covering of spanning trees

Abstract

Designing networks in which every processor has a given number of connections often leads to graphic degree sequence realization models. A nonincreasing sequence d=(d1,d2,…,dn) is graphic if there is a simple graph G with degree sequence d. The spanning tree packing number of graph G, denoted by τ(G), is the maximum number of edge-disjoint spanning trees in G. The arboricity of graph G, denoted by a(G), is the minimum number of spanning trees whose union covers E(G). In this paper, it is proved that, given a graphic sequence d=d1≥d2≥⋯≥dn and integers k2≥k1>0, there exists a simple graph G with degree sequence d satisfying k1≤τ(G)≤a(G)≤k2 if and only if dn≥k1 and 2k1(n−1)≤∑i=1ndi≤2k2(n−|I|−1)+2∑i∈Idi, where I={i:di<k2}. As corollaries, for any integer k>0, we obtain a characterization of graphic sequences with at least one realization G satisfying a(G)≤k, and a characterization of graphic sequences with at least one realization G satisfying τ(G)=a(G)=k.