L(p, q)-Labeling and Integer Flow on Planar Graphs

Given a graph G and two non-negative integers p and q, an L(p, q)-labeling c of G is an assignment of non-negative integers to the vertices of G such that, for any two vertices u and v, |c(u) − c(v)| ≥ p if d(u, v)=1 and |c(u) − c(v)| ≥ q if d(u, v)=2. The L(p, q)-labeling problem arises from a variation of the channel assignment problem. We establish a connection between the L(p, q)-labeling of the planar graph G and the integer flow on the dual graph of G. This provides us with an alternative and potentially more effective way to minimize the edge span of L(p, q)-labelings for planar graphs by using a graph flow approach. As examples, we apply this approach to determine the minimum edge spans of the L(p, q)-labelings for some typical finite planar lattices, including the square lattice, triangular lattice, hexagonal lattice as well as some other lattices consisting of various regular polygons, some of which arise from the design of planar regions for cellular phone networks. Our flow-based approach is also expected to have some further applications in optimizing the other measures for the L(p, q)-labeling problem.