Ted Dobson (Mississippi State University, Starkville, ZDA): On Isomorphisms of vertex-transitive digraphs and homogeneous factorizations

Povzetek: A solving set for a Cayley digraph $\Gamma$ of a group $G$ is a set $S$ such that if $\Gamma'$ is another Cayley digraph of $G$, then $\Gamma$ is isomorphic to $\Gamma'$ if and only if $s(\Gamma) = \Gamma'$ for some $s\in S$. We generalize this notion to non-Cayley digraphs $\Gamma$, and determine necessary and sufficient conditions for a set $S$ to be a solving set of $\Gamma$. A homogeneous factorization of a digraph $\Gamma$ consists of transitive subgroups $M < G\le \Aut(\Gamma)$ and a partition ${\cal P}$ of the edge set of $\Gamma$ such that each cell in the partition is invariant under $M$ and $G$ acts transitively on ${\cal P}$. Using the above result, we show that every homogeneous factorization of $\Gamma$ has a certain "nice" form, which allows for the construction of all homogeneous factorizations of certain types, and all homogeneous factorizations of all vertex-transitive digraphs of some orders.