| Abstract: |
For a graph G and for each vertex v ∈ V (G), let Λ[sub G](v)= {EG(v, 1),E[sub G](v, 2),..., E[sub G](v, κ[sub v])} be a partition of the edges incident with v. Let ΛG = {ΛG(v) | v ∈ V (G)}. We call the pair (G, ΛG) a partitioned graph. Let κ = maxv κv and let g, f : V (G) x {1,..., κ} → ℕ and t, u : V (G) → ℕ be functions where, for all vertices v ∈ V (G), (i) g(v, i) ≤ f(v, i) ≤ dG(v, i) i = 1,..., κv, (ii) ... and ... . A subgraph H of the partitioned graph is said to be a (g, f, u, t)-factor if all vertices v ∈ V (G) satisfy(a) g(v, i) ≤ dH(v, i) ≤ f(v, i),i =1,..., κv and (b) u(v) ≤ dH (v) ≤ t(v), where dH(v, i)= |E(H) ∩ EG(v, i)|. In this paper, we shall show via a reduction to a matching problem, that there is a good algorithm for determining whether a partitioned graph has a (g, f, u, t)-factor. Second, we shall also prove a theorem which characterizes the existence of (0, f, t, u)-factors in a partitioned graph when u(v)
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