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 ω̂ + log² n)-bit labels is presented for general n-vertex graphs with maximum (integral) capacity ω̂. This is shown to be asymptotically optimal. For edge-connectivity, this yields a tight bound of ⊖(log² 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 Ω(k log n) is established for k-vertex connectivity on n-vertex graphs, where k is polylogarithmic in n.