On the factorability of infinite graphs

Abstract

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 $A=BC$. Recently, Maghsoudi et al. proved that the complete graph $K_n$ admits a factorization if and only if $n=4k+1$. 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 $n\ge 2$.