Labeling schemes for flow and connectivity

This paper studies labeling schemes for flow and connectivity functions. A flow labeling scheme using <i>O</i>(log <i>n</i> ṡ log <i>ŵ</i>)-bit labels is presented for general <i>n</i>-vertex graphs with maximum (integral) capacity <i>ŵ</i>. This is shown to be asymptotically optimal. For edge-connectivity, this yields a tight bound of Θ(log² <i>n</i>) bits. A <i>k</i>-vertex connectivity labeling scheme is then given for general <i>n</i>-vertex graphs using at most 3 log <i>n</i> bits for <i>k</i> = 2, 5 log <i>n</i> bits for <i>k</i> = 3 and 2^<i>k</i> log <i>n</i> bits for <i>k</i> > 3. Finally, a lower bound of Ω(<i>k</i> log <i>n</i>) is established for <i>k</i>-vertex connectivity on <i>n</i>-vertex graphs where <i>k</i> is polylogarithmic in <i>n.</i>