Unique Prime Cartesian Factorization of Graphs Over Finite Fields

Abstract

A fundamental result, due to Sabidussi and Vizing, states that every connected graph has a unique prime factorization rel- ative to the Cartesian product; but disconnected graphs are not uniquely prime factorable. This paper describes a system of modular arithmetic on graphs under which both connected and disconnected graphs have unique prime Cartesian factorizations. The Cartesian product of two simple graphs G = (V (G); E(G)) and H = (V (H); E(H)) is the graph G H with V (G H) = V (G) V (H), and (u; x)(v; y) 2 E(G H) if either u = v and xy 2 E(H), or uv 2 E(G) and x = y. This product is commutative and associative: G H = H G and G (H K) = (G H) K (up to isomorphism) for all graphs G, H and K. Also G H is connected if and only if both G and H are connected. For a full treatment of this product, see Chapter 4 of Imrich and Kla zar (2). We denote the empty graph (i.e. the graph with no vertices) as O, and the complete graph on n vertices as Kn. Notice that G O = O and G K1 = G for all graphs G. If n 2 N, then nG denotes the graph that is the disjoint union of n copies of G (or O if n = 0). Note n(G H) = nG H = G nH. For a positive integer n, we dene G n = G G G (n factors) and we adopt the convention G 0 = K1. A graph G is prime if it is nontrivial and G = G1 G2 implies G1 = K1 or G2 = K1. Every graph G has a prime factorization G = G1 G2 Gp, where each factor Gi is prime. A fundamental theorem, proved indepen- dently by Sabidussi (3) and Vizing (4) states that the prime factorization of a connected graph is unique, that is if a connected graph G has prime factor- izations G1 G2 Gp and H1 H2 Hq, then p = q and Gi = Hi for 1 i p (after reindexing, if necessary). But disconnected graphs are not uniquely prime factorable, in general. One standard example is the graph G = K1 + K2 + K 2