L(2,1) L ( 2 , 1 ) -labelings of Cartesian products of two cycles
An L(2,1) L ( 2 , 1 ) -labeling of a graph is an assignment of nonnegative integers to its vertices so that adjacent vertices get labels at least two apart and vertices at distance two get distinct labels. The ? ? -number of a graph G , denoted by ?(G) ? ( G ) , is the minimum range of labels taken over all of its L(2,1) L ( 2 , 1 ) -labelings. We show that the ? ? -number of the Cartesian product of any two cycles is 6, 7 or 8. In addition, we provide complete characterizations for the products of two cycles with ? ? -number exactly equal to each one of these values.
1 ) 1 ) l(2 1) l ( 2 1 ) -labeling l(2 1 ) l(2 1 ) l ( 2 1 ) -coloring l(2 product of cycles vertex labeling distance two labeling