The L(2, 1)-labeling on the skew and converse skew products of graphs

Abstract An L ( 2 , 1 ) -labeling of a graph G is a function f from the vertex set V ( G ) to the set of all nonnegative integers such that | f ( x ) − f ( y ) | ≥ 2 if d ( x , y ) = 1 and | f ( x ) − f ( y ) | ≥ 1 if d ( x , y ) = 2 , where d ( x , y ) denotes the distance between x and y in G . The L ( 2 , 1 ) -labeling number λ ( G ) of G is the smallest number k such that G has an L ( 2 , 1 ) -labeling with max { f ( v ) : v ∈ V ( G ) } = k . Griggs and Yeh conjecture that λ ( G ) ≤ Δ 2 for any simple graph with maximum degree Δ ≥ 2 . This work considers the graph formed by the skew product and the converse skew product of two graphs. As corollaries, the new graph satisfies the above conjecture (with minor exceptions).