Retracts of Infinite Hamming Graphs

A Hamming graph is a Cartesian product of complete graphs. We show that (finite or infinite) quasi-median graphs, which are a generalization of median graphs, are exactly the retracts of Hamming graphs. This generalizes a result of Bandelt (1984,J. Graph Theory8, 501–510) about median graphs, and improves a result of Wilkeit (1992,Discrete Math.102, 191–218) and of Chung, Graham, and Saks (1989,Combinatorica9, 111–131) about finite quasi-median graphs.