On the factorability of infinite graphs

A graph G is said to be factorized into graphs H and K via a matrix product if there exist adjacency matrices A , B , and C for G , H , and K , respectively, such that A = B C . Recently, Maghsoudi et al. proved that the complete graph K n admits a factorization if and only if n = 4 k + 1 . In this note, we show that, in contrast to the finite case, the (countably) infinite complete graph admits a factorization via a matrix product. In addition, they showed that the complete bipartite graph K m , n has a factorization if and only if both m and n are even. We extend this result to the infinite complete bipartite graph K m , n , namely: the infinite complete bipartite graph admits a factorization if and only if the size of the finite part (if it exists) is even. Finally, we show that the n -dimensional grid admits a factorization for all n ≥ 2 .