The diagonal groups are a family of permutation groups defined by a dimension (which we assume to be at least 2) and a group T, which may be finite or infinite and is not required to be simple.
The diagonal graphs are a family of graphs whose automorphism groups are the diagonal groups. They include as special cases two classical families of graphs, the Latin square graphs associated with the Cayley tables of groups, and the distance-transitive folded cubes. I will describe some properties of these graphs: spectrum, clique number, chromatic number, etc.
The determination of the chromatic number is not complete in all cases. Some of the results depend on the truth of the Hall--Paige conjecture (proved by Wilcox, Evans and Bray) on complete mappings of groups. I will describe this and explain how it is used in the analysis.
If time permits, I may describe the extension of the notion of complete mapping to semigroups, and some results on which semigroups admit complete mappings.