The pseudograph [formula omitted]-threshold number

Abstract

For d≥1,s≥0, a (d,d+s)-graph is a graph whose degrees all lie in the interval {d,d+1,…,d+s}. For r≥1,a≥0, an (r,r+a)-factor of a graph G is a spanning (r,r+a)-subgraph of G. An (r,r+a)-factorization of a graph G is a decomposition of G into edge-disjoint (r,r+a)-factors. A pseudograph is a graph which may have multiple edges and may have multiple loops. A loop counts two towards the degree of the vertex it is on. A multigraph here has no loops.For t≥1, let π(r,s,a,t) be the least integer such that, if d≥π(r,s,a,t) then every (d,d+s)-pseudograph G has an (r,r+a)-factorization into x(r,r+a)-factors for at least t different values of x. We call π(r,s,a,t) the pseudograph(r,s,a,t)-threshold number. Let μ(r,s,a,t) be the analogous integer for multigraphs. We call μ(r,s,a,t) the multigraph(r,s,a,t)-threshold number. A simple graph has at most one edge between any two vertices and has no loops. We let σ(r,s,a,t) be the analogous integer for simple graphs. We call σ(r,s,a,t) the simple graph(r,s,a,t)-threshold number.In this paper we give the precise value of the pseudograph π(r,s,a,t)-threshold number for each value of r,s,a and t. We also use this to give good bounds for the analogous simple graph and multigraph threshold numbers σ(r,s,a,t) and μ(r,s,a,t).