Let G be a connected graph with odd girth 2κ + 1. Then G is a (2κ + 1)-angulated graph if every two vertices of G are connected by a path such that each edge of the path is in some (2κ + 1)-cycle. We prove that if G is (2κ + 1 )-angulated, and H is connected with odd girth at least 2κ + 3, then any retract of the box (or Cartesian) product G□H is S□T where S is a retract of G and T is a connected subgraph of H. A graph G is strongly (2κ + 1)-angulated if any two vertices of G are connected by a sequence of (2κ + 1)-cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2κ + 1)-angulated, and H is connected with odd girth at least 2κ+ 1, then any retract of G□H is S□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 (2κ + 1 )-angulated cores must be either one of the factors or the box product itself. Furthermore, if G is a strongly (2κ + 1 )-angulated core, then either Gn is a core for all positive integers n, or the core of Gn 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 (2κ + 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 Gn is a core for any positive integer n. On the other hand, let Gi, be a (2κi, + 1)angulated core for 1 ≤ i ≤ n where κ1 < κ2 < ⋯ < κn. If Gi 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 ≤ i ≤ n - 1, then □i=1n Gi 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)□C 2l+1 is a core for any integers l ≥ r ≥ 2. It is open whether K(r, 2r+ 1)□C2l+1 is a core for r > l > 2.
All Science Journal Classification (ASJC) codes
- Geometry and Topology