Degree multiplicities and independent sets in K_4-free graphs

摘要

Erds et al. [6] asked for what values of k there is a sequence of graphs (Gn)n = 1 such that Gn has n vertices and independence number o(n), contains no K4, and no k + 1 vertices have the same degree. In [6] it is proved that such a k has to be at least 4, but the problem whether any such k exists is left open. By making use of some graphs constructed by Bollobás and Erds [3], we shall prove that for k = 5 (and so for k = 5) there is such a sequence (Gn)n = 1.