Pseudoachromatic and connected-pseudoachromatic indices of the complete graph

TL;DR

This paper investigates the pseudoachromatic and connected-pseudoachromatic indices of complete graphs, providing improved bounds and showing that these indices grow approximately as n^{3/2} for large n.

Contribution

The authors improve the lower bound for the connected pseudoachromatic index and establish upper bounds, demonstrating that these indices are on the order of n^{3/2} for complete graphs.

Findings

Connected pseudoachromatic index of K_n is on the order of n^{3/2}.

Improved lower bounds for the pseudoachromatic index when n=q^2+q+1 with q a power of 2.

Provided upper bounds matching the order of growth for these indices.

Abstract

A complete $k$-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 $\psi'(G)$, 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 $\psi_c'(G)$, 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 $K_n$. Our main contribution is to improve the linear lower bound for the connected pseudoachromatic index given by Abrams and Berman [Australas J Combin 60 (2014), 314--324] and provide an upper bound. These two bounds prove that for any integer $n\geq 8$ the order of $\psi_c'(K_n)$ is $n^{3/2}$. Related to the pseudoachromatic index we prove that for $q$ a power of $2$ and $n=q^2+q+1$, $\psi'(K_n)$ is at least $q^3+2q-3$ which improves the bound $q^3+q$ given by Araujo, Montellano and Strausz [J Graph Theory 66 (2011), 89--97].