Labeling bipartite permutation graphs with a condition at distance two

Abstract

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 , … , λ } 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 λ for which G has L ( p , q ) -labeling is denoted by λ 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 ( 2 p − 1 ) + q ( b c ( G ) − 2 ) , where b c ( G ) is the biclique number of G . Since λ p , q ( G ) ≥ p + q ( b c ( G ) − 2 ) for any bipartite graph G , the upper bound is at most p − 1 far from optimal.