Labeling schemes for flow and connectivity

This paper studies labeling schemes for flow and connectivity functions. A flow labeling scheme using O(log n . log (omega) over cap + log(2) n)-bit labels is presented for general n-vertex graphs with maximum (integral) capacity (omega) over cap. This is shown to be asymptotically optimal. For edge-connectivity, this yields a tight bound of Theta(log(2) n) bits. A k-vertex connectivity labeling scheme is then given for general n-vertex graphs using at most 3 log n bits for k = 2, 5 log n bits for k = 3, and 2(k) log n bits for k > 3. Finally, a lower bound of Omega(k log n) is established for k-vertex connectivity on n-vertex graphs, where k is polylogarithmic in n.