On Grundy indices for complete geometric graphs

TL;DR

This work extends Grundy and pseudo-Grundy coloring notions to complete geometric graphs under several edge-adjacency criteria, analyzing both convex and general-position vertex sets. It provides constructive colorings and counting-based bounds that show quadratic growth in the relevant indices, with geometry-dependent constants and regime-specific results. The methods include circulant decompositions, transversal-convexity results, halving-line techniques, and triangle decompositions, linking Grundy colorings to classical geometric concepts such as thrackles and crossing numbers. The findings enrich the understanding of Grundy-type colorings in geometric contexts and offer techniques potentially applicable to related geometric coloring problems and crossing-number studies.

Abstract

The pseudo-Grundy index of a graph is the largest number of colors that can be assigned to its edges, such that for every pair of colors $i,j$, if $i < j$ then every edge colored with color $j$ is adjacent to at least one edge colored with color $i$. This index has 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 pseudo-Grundy index for geometric graphs, and present results for complete geometric graphs.

Figures (4)

• Figure 1: The coloring of the first two circulants of the complete convex graph with $13$ vertices.
• Figure 2: A set of edges with the same color.
• Figure 3: A chromatic class of $\mathsf{K^{c}_{32}}$
• Figure 4: Configurations of the color classes of size one arising from $F$ and a dashed triangle of $K \setminus F$ in the proof of Theorem \ref{['teo:remainder']}.

Theorems & Definitions (27)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Corollary 1