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Error code: DatasetGenerationError
Exception: UnicodeDecodeError
Message: 'utf-8' codec can't decode byte 0xa9 in position 237: invalid start byte
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1815, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/text/text.py", line 73, in _generate_tables
batch = f.read(self.config.chunksize)
File "/usr/local/lib/python3.9/codecs.py", line 322, in decode
(result, consumed) = self._buffer_decode(data, self.errors, final)
UnicodeDecodeError: 'utf-8' codec can't decode byte 0xa9 in position 237: invalid start byte
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1456, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1055, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 894, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 970, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1702, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1858, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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==== Front
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Front Chem
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Front Chem
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Front. Chem.
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Frontiers in Chemistry
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2296-2646
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Frontiers Media S.A.
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1132587
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10.3389/fchem.2023.1132587
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Chemistry
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Original Research
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Anti-Kekulé number of the {(3, 4), 4}-fullerene*
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Yang and Jia
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10.3389/fchem.2023.1132587
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Yang Rui *
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Jia Huimin
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School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan, China
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Edited by: Baoyindureng Wu, Xinjiang University, China
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Reviewed by: Guifu Su, Beijing University of Chemical Technology, China
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Qiuli Li, Lanzhou University, China
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Hong Bian, Xinjiang Normal University, China
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*Correspondence: Rui Yang, yangrui@hpu.edu.cn
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This article was submitted to Theoretical and Computational Chemistry, a section of the journal Frontiers in Chemistry
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24 2 2023
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2023
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11 113258727 12 2022
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06 2 2023
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Copyright © 2023 Yang and Jia.
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2023
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Yang and Jia
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https://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
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A {(3,4),4}-fullerene graph G is a 4-regular plane graph with exactly eight triangular faces and other quadrangular faces. An edge subset S of G is called an anti-Kekulé set, if G − S is a connected subgraph without perfect matchings. The anti-Kekulé number of G is the smallest cardinality of anti-Kekulé sets and is denoted by akG . In this paper, we show that 4≤akG≤5 ; at the same time, we determine that the {(3, 4), 4}-fullerene graph with anti-Kekulé number 4 consists of two kinds of graphs: one of which is the graph H1 consisting of the tubular graph Qnn≥0 , where Q n is composed of nn≥0 concentric layers of quadrangles, capped on each end by a cap formed by four triangles which share a common vertex (see Figure 2 for the graph Q n ); and the other is the graph H2 , which contains four diamonds D 1, D 2, D 3, and D 4, where each diamond Di1≤i≤4 consists of two adjacent triangles with a common edge ei1≤i≤4 such that four edges e 1, e 2, e 3, and e 4 form a matching (see Figure 7D for the four diamonds D 1 − D 4). As a consequence, we prove that if G∈H1 , then akG=4 ; moreover, if G∈H2 , we give the condition to judge that the anti-Kekulé number of graph G is 4 or 5.
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anti-Kekulé set
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anti-Kekulé number
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{(3,4),4}-fullerene
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perfect matching
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matching
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National Natural Science Foundation of China 10.13039/501100001809 11801148 11626089 This work was supported by the National Natural Science Foundation of China (grant nos. 11801148 and 11626089) and the Foundation for the Doctor of Henan Polytechnic University (grant no. B2014-060).
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==== Body
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pmc1 Introduction
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A {(3,4),4}-fullerene graph G is a 4-regular plane graph with exactly eight triangular faces and other quadrangular faces. This concept of the {(3, 4), 4}-fullerene comes from Deza’s {(R,k)}-fullerene (Deza and Sikirić, 2012). Fixing R ⊂ N, a {(R, k)}-fullerene graph is a k-regular k≥3 , and it is mapped on a sphere whose faces are i-gons i∈R . A {(a,b),k}-fullerene is {(R, k)}-fullerene with R=a,b1≤a≤b . The {(a, b), k}-fullerene draws attention because it includes the mostly widely researched graphs, such as fullerenes (i.e.,{(5, 6), 3}-fullerenes), boron–nitrogen fullerenes (i.e.,{(4, 6), 3}-fullerenes), and (3,6)-fullerenes (i.e.,{(3, 6), 3}-fullerenes) (Yang and Zhang, 2012).
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The anti-Kekulé number of a graph was introduced by Vukičević and Trinajstić (2007). They introduced the anti-Kekulé number as the smallest number of edges that have to be removed from a benzenoid to remain connected but without a Kekulé structure. Here, a Kekulé structure corresponds to a perfect matching in mathematics; it is known that benzenoid hydrocarbon has better stability if it has a lower anti-Kekulé number. Veljan and Vukičević (2008) found that the anti-Kekulé numbers of the infinite triangular, rectangular, and hexagonal grids are 9, 6 and 4, respectively. Zhang et al. (2011) proved that the anti-Kekulé number of cata-condensed phenylenes is 3. For fullerenes, Vukičević (2007) proved that C 60 has anti-Kekulé number 4, and Kutnar et al. (2009) showed that the leapfrog fullerenes have the anti-Kekulé number 3 or 4 and that for each leapfrog fullerene, the anti-Kekulé number can be established by observing the finite number of cases independent of the size of the fullerene. Furthermore, this result was improved by Yang et al. (2012) by proving that all fullerenes have anti-Kekulé number 4.
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In general, Li et al. (2019) showed that the anti-Kekulé number of a 2-connected cubic graph is either 3 or 4; moreover, all (4,6)-fullerenes have the anti-Kekulé number 4, and all the (3,6)-fullerenes have anti-Kekulé number 3. Zhao and Zhang (2020) confirmed all (4,5,6)-fullerenes have anti-Kekulé number 3, which consist of four sporadic (4,5,6)-fullerenes (F 12, F 14, F 18, and F 20) and three classes of (4,5,6)-fullerenes with at least two and at most six pentagons.
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Here, we consider the {(3, 4), 4}-fullerene graphs. In the next section, we recall some concepts and results needed for our discussion. In Section 3, by using Tutte’s Theorem on perfect matching of graphs, we determine the scope of the anti-Kekulé number of the {(3, 4), 4}-fullerene. Finally, we show that the {(3, 4), 4}-fullerene with anti-Kekulé number 4 consists of two kinds of graphs H1,H2 . As a consequence, we prove that if G∈H1 , then akG=4 . Moreover, if G∈H2 , we give the condition to judge that the anti-Kekulé number of graph G is 4 or 5.
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2 Definitions and preliminary results
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Let G=V,E be a simple and connected plane graph with vertex set V(G) and edge set E(G). For V′⊆VG , G − V′ denotes the subgraph obtained from G by deleting the vertices in V′ together with their incident edges. If V′ = v, we write G − v. Similarly, for E′⊆EG , G − E′ denotes the graph with vertex set V(G) and edge set EG−E′ . If E′ = e, we write G − e. Let V′ be a non-empty set; GV′ denotes the induced subgraph of G induced by the vertices of V′; similarly, if E′⊆EG , GE′ denotes the induced subgraph of G induced by the edges of E′.
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For a subgraph H of G, the induced subgraph of G induced by vertices of VG−VH is denoted by H¯ . A plane graph G partitions the rest of the plane into a number of arcwise-connected open sets. These sets are called the faces of G. A face is said to be incident with the vertices and edges in its boundary, and two faces are adjacent if their boundaries have an edge in common. Let FG be the set of the faces of G.
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An edge-cut of a connected plane graph G is a subset of edges C⊆EG such that G − C is disconnected. A k -edge-cut is an edge-cut with k edges. A graph G is k -edge-connected if G cannot be separated into at least two components by removing less than k edges. An edge-cut C of a graph G is cyclic if its removal separates two cycles. A graph G is cyclically k -edge-connected if G cannot be separated into at least two components, each containing a cycle, by removing less than k edges. A cycle is called a facial cycle if it is the boundary of a face.
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For subgraphs H 1 and H 2 of a plane graph G, EH1,H2=EVH1,VH2 represents the set of edges whose two end vertices are in VH1 and VH2 separately. If VH1 and VH2 are two non-empty disjoint vertex subsets such that VH1∪VH2=VG , then EH1,H2 is an edge-cut of G, and we simply write ∇H1=∇VH1 or ∇H2=∇VH2 . We use ∂G to denote the boundary of G, that is, the boundary of the infinite face of G.
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A matching M of a graph G is a set of edges of G such that no two edges from M have a vertex in common. A matching M is perfect if it covers every vertex of G. A perfect matching is also called a Kekulé structure in chemistry.
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Let G be a connected graph with at least one perfect matching. For S⊆EG , we call S an anti-Kekulé set if G − S is connected but has no perfect matchings. The smallest cardinality of anti-Kekulé sets of G is called the anti-Kekulé number and denoted by akG .
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For the edge connectivity of the {(3, 4), 4}-fullerene, we have the following results.
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Lemma 2.1 ((Yang et al., 2023) Lemma 2.3) Every {(3, 4), 4}-fullerene is cyclically 4-edge-connected.
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Lemma 2.2 ((Yang et al., 2023) Corollary 2.4) Every {(3, 4), 4}-fullerene is 4-edge-connected.
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Q n is the graph consisting of n concentric layers of quadrangles, capped on each end by a cap formed by four triangles which share a common vertex as shown in Figure 2. In particular, Q 0 is what we call an octahedron (see Figure 5F).
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Lemma 2.3 ((Yang et al., 2023) Lemma 2.5) If G has a cyclical 4-edge-cut E=e1,e2,e3,e4 , then G≅Qnn≥1 , where the four edges e 1 , e 2 , e 3 , and e 4 form a matching, and each e i belongs to the intersection of two quadrilateral faces for i = 1, 2, 3, 4.
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Tutte’s theorem plays an important role in the process of proof.
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Theorem 2.4 (Lovász and Plummer, 2009) (Tutte’s theorem) A graph G has a perfect matching if and only if for any X⊆VG , oG−X≤X , where oG−X denotes the number of odd components of G − X .
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Here, an odd component of G − X is trivial if it is just a single vertex and non-trivial otherwise.
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All graph-theoretical terms and concepts used but unexplained in this article are standard and can be found in many textbooks, such as Lovász and Plummer (2009).
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3 Main results
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From now on, let G always be a {(3, 4), 4}-fullerene; we called a 4-edge-cut E in G trivial if E=∇v , that is, E consists of the four edges incident to v. By Lemma 2.3, if E is a cyclical 4-edge-cut, then the four edges in E form a matching. Moreover, if E is not a cyclical 4-edge-cut, then E is trivial. So, we have the following lemma.
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Lemma 3.1 Let G be a {(3, 4), 4}-fullerene, E=e1,e2,e3,e4 be an 4-edge-cut, but it is not cyclical, then E is trivial.
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Proof Since E=e1,e2,e3,e4 is an 4-edge-cut, G − E is not connected. Then, G − E has at least two components. Moreover, as G is 4-edge-connected by Lemma 2.2, G − E has at most two components. So, G − E has exactly two components.
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Let G 1, G 2 be two components of G − E. Since E is not cyclical, without loss of generality, we suppose that G 1 is a forest; then, we have n−e=l, (1)
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