Edge Disjoint Spanning Trees ∗

Let Zm be the cyclic group of order m ≥ 3. A graph G is Zm-connected if G has an orientation D such that for any mapping b : V (G) 7→ Zm with ∑ v∈V (G) b(v) = 0, there exists a mapping f : E(G) 7→ Zm − {0} satisfying ∑ e∈E+ D (v) f(e) − ∑ e∈E− D (v) f(e) = b(v) in Zm for any v ∈ V (G); and a graph G is strongly Zm-connected if, for any mapping θ : V (G) → Zm with ∑ v∈V (G) θ(v) = |E(G)| in Zm, there is an orientation D such that d + D(v) = θ(v) in Zm for each v ∈ V (G). In this paper, we study the relation between Zm-connected graphs and strongly Zm-connected graphs and show that a graph G is Zm-connected if and only if (m− 2)G is strongly Zm-connected, where (m−2)G is the graph obtained from G by replacing each edge in G with m−2 parallel edges. We also show that if G is Zm-connected, then (m − 2)G has m − 1 edge disjoint spanning trees. Those results together with a result by Jaeger et al. [J. Combin. Theory Ser. B, 56 (1992), pp. 165–182] imply that every Z3-connected graph is A-connected for any abelian group A with |A| ≥ 4. They are applied to determine the exact values of ex(n,Zm) for all m ≥ 3, where ex(n,Zm) is the largest integer such that every simple graph on n vertices with at most ex(n,Zm) edges is not Zm-connected, and to present characterizations of graphic and multigraphic sequences that have Zm-connected realizations.