On Circular Flows Of Graphs

A sufficient condition for graphs with circular flow index less than 4 is found in this paper. In particular, we give a simple proof of a result obtained by Galluccio and Goddyn (Combinatorica, 2002), and obtain a larger family of such graphs. We refer readers to [1], [2] and [7] for the standard terminology and notations in this paper. The following theorem was proved by Galluccio and Goddyn. Theorem 1 (Galluccio and Goddyn [2]) Let G be a 6-edge-connected graph. Then the circular flow index of G, φC(G) < 4. Here, we give a simple proof of this theorem without using linear programming. Proof. Since G is 6-edge-connected, by Tutte [5]-Theorem 1 or Nash-Williams [4]-Theorem 1, let T1, T2, T3 be three edge disjoint spanning trees of G. Let Pi be a parity subgraph of Ti (for i = 1, 2 only). Now fixing some orientation of G. Since P1 ∪ P2 and G \ E(P1) are even graphs, let f1 be a nowhere-zero 2-flow with support E(P1) ∪E(P2) and f2 be a be a nowhere-zero 2-flow with support E(G)\E(P1). Then f = f1+2f2 is a nowhere-zero 4-flow of G. Reorient the edges of G such that the resulting correspondent 4-flow f∗ > 0. We will show this is the required orientation. First, this orientation is a strong orientation because it is easy to show that each edge is contained in a directed circuits. For each nonempty proper subset X ⊂ V (G), the edge cut δ(X) contains at least one edge in T3, hence having flow value 2. Therefore 3|δ(X)| ≥ outflow of X = inflow of X ≥ |δ−(X)|, ∗Partially supported by the National Security Agency under Grants MDA904-01-1-0022