### 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 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 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 □_{i=1}^{n} 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.

Original language | English (US) |
---|---|

Pages (from-to) | 24-40 |

Number of pages | 17 |

Journal | Journal of Graph Theory |

Volume | 54 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*54*(1), 24-40. https://doi.org/10.1002/jgt.20191

}

*Journal of Graph Theory*, vol. 54, no. 1, pp. 24-40. https://doi.org/10.1002/jgt.20191

**Retracts of box products with odd-angulated factors.** / Che, Zhongyuan; Collins, Karen L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Retracts of box products with odd-angulated factors

AU - Che, Zhongyuan

AU - Collins, Karen L.

PY - 2007/1/1

Y1 - 2007/1/1

N2 - 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.

AB - 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.

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U2 - 10.1002/jgt.20191

DO - 10.1002/jgt.20191

M3 - Article

AN - SCOPUS:33846553618

VL - 54

SP - 24

EP - 40

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -