Generalized hexagonal systems with each hexagon being resonant. (English) Zbl 0774.05077
A hexagonal system (HS) is a finite 2-connected planar graph in which each interior face is a regular hexagon of side length 1, and a generalized hexagonal system (GHS) is a graph obtained by deleting some interior vertices and interior edges from a HS. An edge of a GHS is said to be not fixed if it belongs to some but not all perfect matchings of the GHS. The main result of the paper is the following: A GHS has no fixed edge if and only if the boundaries of its infinite face and nonhexagon faces are alternating cycles. Moreover, if a GHS has no fixed edge then every face of it is an alternating cycle. Here a cycle is called alternating if it is alternating with respect to some perfect matching.
Reviewer: K.Engel (Rostock)
MSC:
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
| 05C38 | Paths and cycles |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
References:
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| [2] | Lovasz, L.; Plummer, M. D., Matching Theory (1986), North-Holland: North-Holland Amsterdam · Zbl 0618.05001 |
| [3] | Zhang, F.; Chen, R., Research on some topological properties of benzenoid systems, Nature J., 10, 3 (1987), (in Chinese). |
| [4] | Zhang, F.; Chen, R., When each hexagon of a hexagonal system covers it, Discrete Appl. Math., 30, 63-75 (1991) · Zbl 0763.05087 |
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