  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 $\chi'([G,P])$ 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 Erd\H{o}s-Faber-Lov\'asz states that every decomposition $[K_n,P]$ of the
complete graph $K_n$ satisfies $\chi'([K_n,P])\leq n$. 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 general position. We also
consider the particular case in which the vertices are in convex position and
present bounds for the chromatic index of a few types of decompositions.
