TITLE: Is the Petersen graph a "Cayley graph" of some more general group-like structure?
QUESTION [12 upvotes]: The Petersen graph is the smallest vertex-transitive graph which is not a Cayley graph. Is it the "Cayley graph" of some slightly more general group-like structure?

REPLY [9 votes]: We are working on these questions currently with Ignacio Garcia-Marco. As a postive answer I can tell you that the multiplication table of S codes a semigroup. If you take elements 1,6 as connection set, then the right Cayley graph is Cay(S,{1,6}). Its underlying undirected simple graphs is the Petersen graph.
From the negative side we can also show, that any directed Cayley graph (of a semigroup) whose underlying undirected simple graph is the Petersen graph has loops.