The f-factor Problem for Graphs and the Hereditary Property

If P is a hereditary property then we show that, for the existence of a perfect f -factor, P is a sufficient condition for countable graphs and yields a sufficient condition for graphs of size א1. Further we give two examples of a hereditary property which is even necessary for the existence of a perfect f -factor. We also discuss the א2-case. We consider graphs G = (V,E), where V = V (G) is a nonempty set of vertices and E = E(G) ⊆ { e ⊆ V : |e| = 2} is the set of edges of G. If x is a vertex of G and F ⊆ E, then we denote by dF (x) the cardinal |{e ∈ F : x ∈ e}|. dF (x) is called the degree of x with respect to F and dE(x) the degree of x. ON denotes the class of ordinals, CN the class of cardinals. Greek letters α, β, γ, . . . always denote ordinals, whereas the middle letters κ, λ, μ, ν, . . . are reserved for infinite cardinals. Let G = (V,E) be a graph, f : V → CN be a function and F ⊆ E. F is said to be an f-factor of G if dF (x) ≤ f(x) for all x ∈ V . We call an f -factor F of G perfect if dF (x) = f(x) for all x ∈ V . For κ ∈ CN we denote f−1(κ) := {x ∈ V : f(x) = κ}. Let C be the class of all ordered pairs (G, f), such that G = (V,E) is a graph, f : V → CN is a function, and f(x) ≤ dE(x) for all x ∈ V . This paper discusses the problem to find a necessary and sufficient condition for the existence of a perfect f -factor of a graph. In [5], Tutte published a criterion for finite graphs, and in [4] Niedermeyer solved the problem for countable graphs and functions f : V −→ ω. We present a solution for graphs of size א0 and functions f : V −→ ω∪{א0}, a solution for graphs of size א1, and discuss the א2-case. If H ⊆ E, then denote by G − H the graph (V,E \ H), and if e ∈ E, then let G − e be the graph G − {e}. If x, y ∈ V , denote by fx,y : V → CN the function defined by fx,y(v) := { f(v) − 1 if v ∈ {x, y} and 1 ≤ f(v) < א0 f(v) else . Now let P be a formula with two free variables. P (G, f) means that (G, f) ∈ C and (G, f) has the property P . P is said to be hereditary if for every (G, f) with P (G, f), for every vertex x ∈ V (G) with f(x) > 0 there exists a vertex y ∈ V (G) with f(y) > 0, {x, y} ∈ E(G), and P (G − {x, y}, fx,y). ∗This paper was supported by the Volkswagen Stiftung