Coloring decompositions of complete geometric graphs

Abstract

A decomposition of a non-empty simple graph G is a pair [G, P ], such that P is a set of non-empty induced subgraphs of G, and every edge of G belongs to exactly one subgraph in P . The chromatic index χ ([G, P ]) of a decomposition [G, P ] is the smallest number k for which there exists a k-coloring of the elements of P in such a way that: for every element of P all of its edges have the same color, and if two members of P share at least one vertex, then they have different colors. A long standing conjecture of Erdős-Faber-Lovász states that every decomposition [Kn, P ] of the complete graph Kn satisfies χ ([Kn, P ]) ≤ n. 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 index of several types of decompositions when the vertices of the graph are in This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922.general position. We also consider the particular case when the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.

On crossing families of complete geometric graphs

“…K n contains a P 4 -intersecting family of size at least n 2 400 . In [10] it was proven that any K n contains a K 4 -intersecting family of size n 2 /24.5. To conclude, we propose the following two conjectures.…”