Yang Rui, Jia Huimin
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan, China.
Front Chem. 2023 Feb 24;11:1132587. doi: 10.3389/fchem.2023.1132587. eCollection 2023.
A is a 4-regular plane graph with exactly eight triangular faces and other quadrangular faces. An edge subset of is called an , if - is a connected subgraph without perfect matchings. The of is the smallest cardinality of anti-Kekulé sets and is denoted by . In this paper, we show that ; 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 consisting of the tubular graph , where is composed of 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 ); and the other is the graph , which contains four diamonds , , , and , where each diamond consists of two adjacent triangles with a common edge such that four edges , , , and form a matching (see Figure 7D for the four diamonds - ). As a consequence, we prove that if , then ; moreover, if , we give the condition to judge that the anti-Kekulé number of graph is 4 or 5.
A是一个4正则平面图，恰好有八个三角形面和其他四边形面。的一个边子集被称为一个反凯库勒集，如果(G - S)是一个没有完美匹配的连通子图。的反凯库勒数是反凯库勒集的最小基数，用(ak(G))表示。在本文中，我们证明了(ak(G)\geq4)；同时，我们确定反凯库勒数为4的({(3, 4), 4}) - 富勒烯图由两种图组成：一种是由管状图(T_n)构成的图(G_1)，其中(T_n)由(n)个同心四边形层组成，两端由四个共享一个公共顶点的三角形形成的帽封顶（图(G_1)见图2）；另一种是图(G_2)，它包含四个菱形(D_1)、(D_2)、(D_3)和(D_4)，其中每个菱形(D_i)由两个有公共边的相邻三角形组成，使得四条边(e_1)、(e_2)、(e_3)和(e_4)形成一个匹配（四个菱形(D_1 - D_4)见图7D）。因此，我们证明了如果(G)是({(3, 4), 4}) - 富勒烯图，那么(ak(G)\geq4)；此外，如果(G)是({(3, 4), 4}) - 富勒烯图，我们给出判断图(G)的反凯库勒数是4还是5的条件。
