Elsevier

Discrete Applied Mathematics

Volume 231, 20 November 2017, Pages 60-66
Discrete Applied Mathematics

Pseudoachromatic and connected-pseudoachromatic indices of the complete graph

https://doi.org/10.1016/j.dam.2017.03.019Get rights and content
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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 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 n8 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+2q3 which improves the bound q3+q given by Araujo et al. (2011).

Keywords

Hadwiger number
Complete colorings
Pseudoachromatic index
Projective plane
Connected-pseudoachromatic index

Cited by (0)

Research partially supported by CONACyT-México under projects 178395, 166306, and PAPIIT-México under project IN104915 and IN104609-3. The second author is partially supported by a CONACyT-México Postdoctoral Fellowship and by the National Scholarship Program of the Slovak Republic.