An extension of the channel-assignment problem: L(2, 1)-labelings of generalized Petersen graphs

The channel-assignment problem involves assigning frequencies represented by nonnegative integers to radio transmitters such that transmitters in close proximity receive frequencies that are sufficiently far apart to avoid interference. In one of its variations, the problem is commonly quantified as follows: transmitters separated by the smallest unit distance must be assigned frequencies that are at least two apart and transmitters separated by twice the smallest unit distance must be assigned frequencies that are at least one apart. Naturally, this channel-assignment problem can be modeled with vertex labelings of graphs. An L(2, 1)-labeling of a graph G is a function f from the vertex set V(G) to the 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. The λ-number of G, denoted λ(G), is the smallest number k such that G has an L(2, 1)-labeling using integers from {0,1,...,k}. A long-standing conjecture by Griggs and Yeh stating that λ(G) can not exceed the square of the maximum degree of vertices in G has motivated the study of the λ-numbers of particular classes of graphs. This paper provides upper bounds for the λ-numbers of generalized Petersen graphs of orders 6, 7, and 8. The results for orders 7 and 8 establish two cases in a conjecture by Georges and Mauro, while the result for order 6 improves the best known upper bound. Furthermore, this paper provides exact values for the λ-numbers of all generalized Petersen graphs of order 6.