Coloring decompositions of complete geometric graphs
Clemens Huemer, Dolores Lara, and Christian Rubio-Montiel

TL;DR
This paper explores the coloring properties of decompositions of complete geometric graphs, providing bounds on the chromatic index for various configurations, inspired by a classical conjecture in graph theory.
Contribution
It introduces the concept of chromatic index for geometric graph decompositions and establishes bounds for different vertex arrangements, extending classical graph theory conjectures.
Findings
Bounds established for the chromatic index in general position
Bounds established for the convex position case
Extension of the Erdős-Faber-Lovász conjecture to geometric graphs
Abstract
A decomposition of a non-empty simple graph is a pair , such that is a set of non-empty induced subgraphs of , and every edge of belongs to exactly one subgraph in . The chromatic index of a decomposition is the smallest number for which there exists a -coloring of the elements of in such a way that: for every element of all of its edges have the same color, and if two members of share at least one vertex, then they have different colors. A long standing conjecture of Erd\H{o}s-Faber-Lov\'asz states that every decomposition of the complete graph satisfies . In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic…
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