L(2, 1)-labeling of perfect elimination bipartite graphs

Abstract An L ( 2 , 1 ) -labeling of a graph G is an assignment of nonnegative integers, called colors, to the vertices of G such that the difference between the colors assigned to any two adjacent vertices is at least two and the difference between the colors assigned to any two vertices which are at distance two apart is at least one. The span of an L ( 2 , 1 ) -labeling f is the maximum color number that has been assigned to a vertex of G by f . The L ( 2 , 1 ) -labeling number of a graph G , denoted as λ ( G ) , is the least integer k such that G has an L ( 2 , 1 ) -labeling of span k . In this paper, we propose a linear time algorithm to L ( 2 , 1 ) -label a chain graph optimally. We present constant approximation L ( 2 , 1 ) -labeling algorithms for various subclasses of chordal bipartite graphs. We show that λ ( G ) = O ( Δ ( G ) ) for a chordal bipartite graph G , where Δ ( G ) is the maximum degree of G . However, we show that there are perfect elimination bipartite graphs having λ = Ω ( Δ 2 ) . Finally, we prove that computing λ ( G ) of a perfect elimination bipartite graph is NP-hard.