Tight products and graph expansion

In this article, 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 G ₁ and G₂, if V(H) = V(G₁) × V(G ₂) 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(d^3/4). This construction resembles random graph lifts, but requires fewer random bits.