L(2,1)-coloring of the Fibonacci cubes

Abstract

An L(2, l)-coloring of a graph G is an assignment of labels from {0,1,..., A} to the vertices of G such that vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. The X-number X(G) of G is the minimum value A such that G admits an L(2,1)-coloring. It is well known that the problem of determining the X-number is NP-hard. The Fibonacci cube network was recently proposed as an alternative to the hypercube network. Three different evolutionary algorithms are presented to find optimal or near optimal L(2,1)-coloring of the Fibonacci cubes. The algorithms are compared with the Petford-Welsh probabilistic algorithm