An Intermediate Value Theorem for the Arboricities

Let G be a graph. The vertex edge arboricity of G denoted by a G a1 G is the minimum number of subsets into which the vertex edge set of G can be partitioned so that each subset induces an acyclic subgraph. Let d be a graphical sequence and let R d be the class of realizations of d. We prove that if π ∈ {a, a1}, then there exist integers x π and y π such that d has a realization G with π G z if and only if z is an integer satisfying x π ≤ z ≤ y π . Thus, for an arbitrary graphical sequence d and π ∈ {a, a1}, the two invariants x π min π,d : min{π G : G ∈ R d } and y π max π,d : max{π G : G ∈ R d } naturally arise and hence π d : {π G : G ∈ R d } {z ∈ Z : x π ≤ z ≤ y π }. We write d r : r, r, . . . , r for the degree sequence of an r-regular graph of order n. We prove that a1 r { r 1 /2 }. We consider the corresponding extremal problem on vertex arboricity and obtain min a, r in all situations and max a, r for all n ≥ 2r 2.