Pseudoachromatic and connected-pseudoachromatic indices of the complete graph

Abstract

A completek-coloring of a graph G is a (not necessarily proper) k-coloring of the vertices of G, such that each pair of different colors appears in an edge. A complete k-coloring is also called connected, if each color class induces a connected subgraph of G. The pseudoachromatic index of a graph G, denoted by ψ′(G), is the largest k for which the line graph of G has a complete k-coloring. Analogously the connected-pseudoachromatic index of G, denoted by ψc′(G), is the largest k for which the line graph of G has a connected and complete k-coloring.In this paper we study these two parameters for the complete graph Kn. Our main contribution is to improve the linear lower bound for the connected pseudoachromatic index given by Abrams and Berman (2014) and provide an upper bound. These two bounds prove that for any integer n≥8 the order of ψc′(Kn) is n3/2.Related to the pseudoachromatic index we prove that for q a power of 2 and n=q2+q+1, ψ′(Kn) is at least q3+2q−3 which improves the bound q3+q given by Araujo et al. (2011).