Tight products and graph expansion

In this paper we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G1 and G2, if V (H) = V (G1) × V (G2) and both projection maps V (H) → V (G1) and V (H) → V (G2) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP -hard to decide). We investigate the conditions under which this is possible. This perspective yields a new characterization of class-1 (2k + 1)-regular graphs. We also obtain a new model of random d-regular graphs whose second eigenvalue is almost surely at most O(d3/4). This construction resembles random graph lifts, but requires fewer random bits.