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

A note on the Erd\"os-Faber-Lov\'asz Conjecture: quasigroups and complete digraphs