| Title: | Kekulé structure of angularly connected even ring systems |
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| Authors: | ID Brezovnik, Simon (Author) |
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| Files: |  PDF - Presentation file, download (315,29 KB)
MD5: 9884C77D6855F3DDD63FE35C14356FDA
 URL - Source URL, visit https://www.mdpi.com/2075-1680/13/12/827
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | An even ring system $G$ 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. $G$ is angularly connected if all of the peripheral segments of $G$ have odd lengths. In this paper, we show that every angularly connected even ring system $G$, 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 $G$, 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 $G$, 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 $A$. 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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| Keywords: | Kekulé structure, Clar structure, perfect matching, benzenoid system, even ring system, face coloring, edge coloring, Clar set |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | str. 1-14 |
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| Numbering: | Vol. 13, iss. 12, [art. no.] 827 |
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| PID: | 20.500.12556/DiRROS-21066  |
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| UDC: | 519.17 |
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| ISSN on article: | 2075-1680 |
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| DOI: | 10.3390/axioms13120827 |
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| COBISS.SI-ID: | 216596995 |
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| Note: |
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| Publication date in DiRROS: | 18.12.2024 |
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| Views: | 32 |
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| Downloads: | 9 |
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