Retracts of Odd-angulated Graphs and Construction of Cores

Let G be a connected graph with odd girth 2k + 1. Then G is a (2k + 1)-angulated graph if every two vertices of G are joined by a path such that each edge of the path is in some (2k + 1)-cycle. We prove that if G is (2k+1)-angulated, and H is connected with odd girth at least 2k+3, then any retract R of the box (or Cartesian) product G2H is isomorphic to S2T where S is a retract of G and T is a subgraph of H. A graph G is strongly (2k + 1)angulated if any two vertices of G are connected by a sequence of (2k+1)-cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k+1)-angulated, and H is connected with odd girth at least 2k+1, then any retract R of the Cartesian (or box product) G2H is isomorphic to S2T where S is a retract of G and T is a subgraph of H or S is a single vertex and T is a retract of H. These two results improve theorems on weakly and strongly triangulated graphs by Nowakowski and Rival in [9]. As a corollary, we get that the core of two strongly (2k+1)-angulated cores must be either one of the factors or the product itself. We construct cores from graphs that have a vertex which is fixed under any of its automorphisms, and also from vertex-transitive graphs. In particular, the box product M(G)2M(G) is a core if M(G) is a core, where M(G) is the graph resulting from the Mycielski construction on G. Further, the box product of any two Kneser graphs K(n, 2n+1)2K(m, 2m+1) is a core whenever n, m ≥ 2; and K(n, 2n+1)2C2m+1 is a core for m ≥ n ≥ 2.