  A decomposition of a simple graph $G$ is a pair $(G,P)$ where $P$ is a set of
subgraphs of $G$, which partitions the edges of $G$ in the sense that every
edge of $G$ belongs to exactly one subgraph in $P$. If the elements of $P$ are
induced subgraphs then the decomposition is denoted by $[G,P]$.
  A $k$-$P$-coloring of a decomposition $(G,P)$ is a surjective function that
assigns to the edges of $G$ a color from a $k$-set of colors, such that all
edges of $H\in P$ have the same color, and, if $H_1,H_2\in P$ with $V(H_1)\cap
V(H_2)\neq\emptyset$ then $E(H_1)$ and $E(H_2)$ have different colors. The
\emph{chromatic index} $\chi'((G,P))$ of a decomposition $(G,P)$ is the
smallest number $k$ for which there exists a $k$-$P$-coloring of $(G,P)$.
  The well-known Erd\"os-Faber-Lov\'asz Conjecture states that any
decomposition $[K_n,P]$ satisfies $\chi'([K_n,P])\leq n$. We use quasigroups
and complete digraphs to give a new family of decompositions that satisfy the
conjecture.

Combinatorics