Fast exact algorithm for L(2,1)-labeling of graphs - Junosza-Szaniawski, Konstanty; Kratochvíl, Jan; Liedloff, Mathieu; Rossmanith, Peter; Rzążewski, Paweł (Theoretical computer science, 2013)
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∗(2.6488n). 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∗(2.5944n) for claw-free graphs, and in time O∗(2n−r(2+nr)r) for graphs having a dominating set of size r.