Retracts of box products with odd-angulated factors

Let G be a connected graph with odd girth 2 kappa + 1. Then G is a (2 kappa + 1)-angulated graph if every two vertices of G are connected by a path such that each edge of the path is in some (2 kappa + 1)-cycle. We prove that if G is (2 kappa + 1)-angulated, and H is connected with odd girth at least 2 kappa + 3, then any retract of the box (or Cartesian) product G square H is S square T where S is a retract of G and T is a connected subgraph of H. A graph G is strongly (2 kappa + 1)-angulated if any two vertices of G are connected by a sequence of (2 kappa + 1)-cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2 kappa + 1)-angulated, and H is connected with odd girth at least 2 kappa + 1, then any retract of G square H is S square T where S is a retract of G and T is a connected subgraph of H or \V(S)\ = 1 and T is a retract of H. These two results improve theorems on weakly and strongly triangulated graphs by Nowakowski and Rival [Disc Math 70 (1988), 169-184]. As a corollary, we get that the core of the box product of two strongly (2 kappa + 1)-angulated cores must be either one of the factors or the box product itself. Furthermore, if G is a strongly (2 kappa + 1)-angulated core, then either G(n) is a core for all positive integers n, or the core of G(n) is G for all positive integers n. In the latter case, G is homomorphically equivalent to a normal Cayley graph [Larose, Laviolette, Tardiff, European J Combin 19 (1998), 867-881]. In particular, let G be a strongly (2 kappa+1)-angulated core such that either G is not vertex transitive, or G is vertex-transitive and any two maximum independent sets have non-empty intersection. Then G(n) is a core for any positive integer n. On the other hand, let G(i) be a (2 kappa(i) + 1)-angulated core for 1 <= i <= n where kappa(1) < kappa(2) <... < kappa(n). If G(i) has a vertex that is fixed under any automorphism for 1 <= i <= n - 1, or G(i) is vertex-transitive such that any two maximum independent sets have non-empty intersection for 1 <= r <= n - 1, then square(n)(i=1) G(i) is a core. We then apply the results to construct cores that are box products with Mycielski construction factors or with odd graph factors. We also show that K(r, 2r+ 1)square C2/+1 is a core for any integers / >= r >= 2. It is open whether K(r, 2r+ 1)square C2/+1 is a core for r > / >= 2. (C) 2006 Wiley Periodicals, Inc.