Abstrakt
An L(2, 1)-labeling of a graph is a mapping from its vertex set into nonnegative integers such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. The span of such a labeling is the maximum label used, and the L(2, 1)-span of a graph is the minimum possible span of its L(2, 1)-labelings.
We show how to compute the L(2, 1)-span of a connected graph in time O*(2.6488(n)). Previously published exact exponential time algorithms were gradually improving the base of the exponential function from 4 to the so far best known 3, with 3 itself seemingly having been the Holy Grail for quite a while.
As concerns special graph classes, we are able to solve the problem in time O*(2.5944(n)) for claw-free graphs, and in time O*(2(n-r)(2 + n/r)(r)) for graphs having a dominating set of size r.