### Abstract

An even cycle C in a graph G is called a conjugated circuit if a perfect matching of C can be extended to a perfect matching of G. An edge of a graph G is called a forcing edge if it is contained in exactly one perfect matching of G. The two concepts have played important roles in the studies of Kekulé structures of benzenoid hydrocarbon molecules. In this paper, we first present a different proof for a result just published in the 2012 article [9] by Klavžar and Salem, that is, all circuits of a 2-connected outerplanar bipartite graph are conjugated. Then we further give a characterization for the conjugated circuits in any 2-connected (no matter whether bipartite or not) outerplanar graph with an even number of vertices. We also show that each edge of a connected bipartite graph G is a forcing edge if and only if G is an even cycle or an edge, which generalizes a main result in the 1991 paper [6] on polyhexes by Harary, Klein, and Živković. Finally we present miscellaneous related results on perfect matching forcings, one of which asserts that a bipartite graph G with 2n vertices has its forcing number attaining the largest possible value n-1 if and only if G is the complete bipartite graph K_{n,n}.

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

Pages (from-to) | 721-732 |

Number of pages | 12 |

Journal | Match |

Volume | 69 |

Issue number | 3 |

State | Published - 2013 |

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### All Science Journal Classification (ASJC) codes

- Chemistry(all)
- Computer Science Applications
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

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**Conjugated circuits and forcing edges.** / Che, Zhongyuan; Chen, Zhibo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Conjugated circuits and forcing edges

AU - Che, Zhongyuan

AU - Chen, Zhibo

PY - 2013

Y1 - 2013

N2 - An even cycle C in a graph G is called a conjugated circuit if a perfect matching of C can be extended to a perfect matching of G. An edge of a graph G is called a forcing edge if it is contained in exactly one perfect matching of G. The two concepts have played important roles in the studies of Kekulé structures of benzenoid hydrocarbon molecules. In this paper, we first present a different proof for a result just published in the 2012 article [9] by Klavžar and Salem, that is, all circuits of a 2-connected outerplanar bipartite graph are conjugated. Then we further give a characterization for the conjugated circuits in any 2-connected (no matter whether bipartite or not) outerplanar graph with an even number of vertices. We also show that each edge of a connected bipartite graph G is a forcing edge if and only if G is an even cycle or an edge, which generalizes a main result in the 1991 paper [6] on polyhexes by Harary, Klein, and Živković. Finally we present miscellaneous related results on perfect matching forcings, one of which asserts that a bipartite graph G with 2n vertices has its forcing number attaining the largest possible value n-1 if and only if G is the complete bipartite graph Kn,n.

AB - An even cycle C in a graph G is called a conjugated circuit if a perfect matching of C can be extended to a perfect matching of G. An edge of a graph G is called a forcing edge if it is contained in exactly one perfect matching of G. The two concepts have played important roles in the studies of Kekulé structures of benzenoid hydrocarbon molecules. In this paper, we first present a different proof for a result just published in the 2012 article [9] by Klavžar and Salem, that is, all circuits of a 2-connected outerplanar bipartite graph are conjugated. Then we further give a characterization for the conjugated circuits in any 2-connected (no matter whether bipartite or not) outerplanar graph with an even number of vertices. We also show that each edge of a connected bipartite graph G is a forcing edge if and only if G is an even cycle or an edge, which generalizes a main result in the 1991 paper [6] on polyhexes by Harary, Klein, and Živković. Finally we present miscellaneous related results on perfect matching forcings, one of which asserts that a bipartite graph G with 2n vertices has its forcing number attaining the largest possible value n-1 if and only if G is the complete bipartite graph Kn,n.

UR - http://www.scopus.com/inward/record.url?scp=84898002996&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84898002996&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84898002996

VL - 69

SP - 721

EP - 732

JO - Match

JF - Match

SN - 0340-6253

IS - 3

ER -