Retracts of box products with odd‐angulated factors

Abstract

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 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 Journal of Graph Theory © 2006 Wiley Periodicals, Inc. 24 RETRACTS OF ODD-ANGULATED GRAPHS 25normal 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 G n is a core for any positive integer n. On the other hand, let G i be a (2κ i + 1)-angulated core for 1 ≤ i ≤ n where κ 1 < κ 2 < · · · < κ 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 ≤ i ≤ n − 1, then 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) C 2l+1 is a core for any integers l ≥ r ≥ 2. It is open whether K(r, 2r + 1) C 2l+1 is a core for r > l ≥ 2.

Odd‐angulated graphs and cancelling factors in box products

“…Note that (A&B) k → (A B) k → A B. However, A B is an induced subgraph of (A&B) 2 , so A B is a retract of (A&B) k for k ≥ 2. Recall that A B is also a core.…”

“…See also Klavžar [6], which corrected some errors in [8]. Recently, Che and Collins [2] extended this result to the box product G H with one factor that is (2k + 1)-angulated or strongly (2k + 1)-angulated. They proved the following result.…”

“…Thus, we ask: By the main theorems in [2], we can see that if G is strongly (2k + 1)-angulated (or, (2k + 1)-angulated ) and H is a connected graph with odd girth at least 2k + 1 (or, 2k + 3, respectively), then the core of G H can be written as the box product of subgraphs of G and H.…”

“…The question was raised in [2] whether the hypothesis that both G and H are strongly (2k + 1)-angulated in Theorem 1.1 could be replaced by the hypothesis that at least one of G and H is (2k + 1)-angulated. The answer is no.…”

“…They also defined a graph to be weakly hom-idempotent if there exists a homomorphism from G n+1 to G n for some n ≥ 1. The Petersen graph is not weakly hom-idempotent [1], W 5 is not weakly hom-idempotent [2,10], and Kneser graphs are not weakly hom-idempotent [7].…”