Hamiltonicity, diameter, domination, packing, and biclique partitions of Mycielski's graphs

Abstract

Let ω( G), x( G), A( G), bp( G), diam( G), η( G), and γ( G) be the clique number, chromatic number, adjacency matrix, biclique partition number, diameter, packing number, and domination number of a connected graph G. Mycielski constructed a graph μ( G) with ω( μ( G)) = ω( G) and x( μ( G)) = x( G) + 1. We show: if G is Hamiltonian, then so is μ( G); if A( G) and A( G + v) ( G + v is G joined with a vertex) are invertible, then so is A( μ( G)) and further bp(μ(G)) = ¦G¦ + 1; η(μ(G)) = η(G); γ(μ(G)) = γ(G) + 1; diam( μ( G)) = min( max(2, diam( G)), 4); and more.