Kekulé Structure of Angularly Connected Even Ring Systems
An even ring system <i>G</i> is a simple 2-connected plane graph with all interior vertices of degree 3, all exterior vertices of either degree 2 or 3, and all finite faces of an even length. <i>G</i> is angularly connected if all of the peripheral segments of <i>G</...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-11-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/12/827 |
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| Summary: | An even ring system <i>G</i> is a simple 2-connected plane graph with all interior vertices of degree 3, all exterior vertices of either degree 2 or 3, and all finite faces of an even length. <i>G</i> is angularly connected if all of the peripheral segments of <i>G</i> have odd lengths. In this paper, we show that every angularly connected even ring system <i>G</i>, which does not contain any triple of altogether-adjacent peripheral faces, has a perfect matching. This was achieved by finding an appropriate edge coloring of <i>G</i>, derived from the proof of the existence of a proper face 3-coloring of the graph. Additionally, an infinite family of graphs that are face 3-colorable has been identified. When interpreted in the context of the inner dual of <i>G</i>, this leads to the introduction of 3-colorable graphs containing cycles of lengths 4 and 6, which is a supplementation of some already known results. Finally, we have investigated the concept of the Clar structure and Clar set within the aforementioned family of graphs. We found that a Clar set of an angularly connected even ring system cannot in general be obtained by minimizing the cardinality of the set <i>A</i>. This result is in contrast to the previously known case for the subfamily of benzenoid systems, which admit a face 3-coloring. Our results open up avenues for further research into the properties of Clar and Fries sets of angularly connected even ring systems. |
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| ISSN: | 2075-1680 |