Geometric achromatic and pseudoachromatic indices

TL;DR

This paper extends the concepts of pseudoachromatic and achromatic indices to geometric graphs, specifically analyzing complete geometric graphs with points in convex position, and provides exact formulas for these indices.

Contribution

It introduces geometric versions of these indices and determines their exact values for complete geometric graphs in convex position.

Findings

Achromatic and pseudoachromatic indices for convex complete geometric graphs are 6a4 6a4 ",

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Abstract

The pseudoachromatic index of a graph is the maximum number of colors that can be assigned to its edges, such that each pair of different colors is incident to a common vertex. If for each vertex its incident edges have different color, then this maximum is known as achromatic index. Both indices have been widely studied. A geometric graph is a graph drawn in the plane such that its vertices are points in general position, and its edges are straight-line segments. In this paper we extend the notion of pseudoachromatic and achromatic indices for geometric graphs, and present results for complete geometric graphs. In particular, we show that for $n$ points in convex position the achromatic index and the pseudoachromatic index of the complete geometric graph are $\lfloor \tfrac{n^2+n}{4} \rfloor$.