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 bba256b54" title="Click to view the MathML source">x(r,r+a)-factors for at least bbae2c" title="Click to view the MathML source">t different values of bba256b54" title="Click to view the MathML source">x. We call π(r,s,a,t) the pseudograph (r,s,a,t)-threshold number . Let ba2319" title="Click to view the MathML source">μ(r,s,a,t) be the analogous integer for multigraphs. We call ba2319" title="Click to view the MathML source">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 bbae2c" title="Click to view the MathML source">t. We also use this to give good bounds for the analogous simple graph and multigraph threshold numbers σ(r,s,a,t) and ba2319" title="Click to view the MathML source">μ(r,s,a,t).