L(2, 1)-Labeling of Kneser graphs and coloring squares of Kneser graphs.

The frequency assignment problem is to assign a frequency to each radio transmitter so that transmitters are assigned frequencies with allowed separations. Motivated by a variation of the frequency assignment problem, the L(2,1)-labeling problem was put forward. 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 (G) of G is the smallest number k such that G has a L(2,1)-labeling with max{f(v):vV(G)}=k. Griggs and Yeh (1992) conjectured that (G)2 for any graph with maximum degree 2. The Kneser graphK(a,b) is defined as the graph whose vertices correspond to all b-subsets of the a-set A={1,2,,a}, with edges joining pairs of vertices that correspond to non-overlapping b-subsets. The chromatic number of the Kneser graph K(a,b) is a2b+2. Fredi put forward an open problem: What is the value for the chromatic number of the square of any Kneser graph K(a,b) (the square of a graph is the graph obtained by adding edges joining vertices at distance 2)? In this article, the L(2,1)-labeling numbers of Kneser graphs K(a,b) are considered. The combined upper bounds for the L(2,1)-labeling numbers of Kneser graphs are derived by using two approaches. The exact L(2,1)-labeling number of Kneser graph K(a,b) for a3b1 is obtained, and it is proved that Griggs and Yehs conjecture holds for Kneser graphs and the L(2,1)-labeling numbers of Kneser graphs are much better than 2 in most cases. We also provide bounds for the chromatic number of the square of any Kneser graph K(a,b) using the proof for the upper bounds of the L(2,1)-labeling numbers of Kneser graphs.