Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,...,@l} such that |f(u)-f(v)|>=p if u and v are adjacent, and |f(u)-f(v)|>=q if u and v are at distance 2 apart. The minimum value of @l for which G has L(p,q)-labeling is denoted by @l"p","q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks. In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p-1)+q(bc(G)-2), where bc(G) is the biclique number of G. Since @l"p","q(G)>=p+q(bc(G)-2) for any bipartite graph G, the upper bound is at most p-1 far from optimal.