The complexity of the -labeling problem for bipartite planar graphs of small degree

Given a simple graph G , by an L ( p , q ) -labeling of G we mean a function c that assigns nonnegative integers to its vertices in such a way that if two vertices u , v are adjacent then | c ( u ) − c ( v ) | ≥ p , and if they are at distance 2 then | c ( u ) − c ( v ) | ≥ q . The L ( p , q ) -labeling problem can be defined as follows: given a graph G and integer t , determine whether there exists an L ( p , q ) -labeling c of G such that c ( V ) ⊆ { 0 , 1 , … , t } . In the paper we show that the problem is N P -complete even when restricted to bipartite planar graphs of small maximum degree and for relatively small values of t . More precisely, we prove that: (1) lt; 3 q then the problem is N P -complete for bipartite planar graphs of maximum degree Δ ≤ 3 and t = p + max { 2 q , p } ; 3 q then the problem is N P -complete for bipartite planar graphs of maximum degree Δ ≤ 4 and t = 6 q ; gt; 3 q then the problem is N P -complete for bipartite planar graphs of maximum degree Δ ≤ 4 and t = p + 5 q . ticular, these results imply that the L ( 2 , 1 ) -labeling problem in planar graphs is N P -complete for t = 4 , and that the L ( p , q ) -labeling problem in graphs of maximum degree Δ ≤ 4 is N P -complete for all values of p and q , thus answering two well-known open questions.