AUTHOR=Yang Rui , Jia Huimin TITLE=Anti-Kekulé number of the {(3, 4), 4}-fullerene* JOURNAL=Frontiers in Chemistry VOLUME=11 YEAR=2023 URL=https://www.frontiersin.org/journals/chemistry/articles/10.3389/fchem.2023.1132587 DOI=10.3389/fchem.2023.1132587 ISSN=2296-2646 ABSTRACT=

A {(3,4),4}-fullerene graphG 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 D₁, D₂, D₃, and D₄, where each diamond Di1≤i≤4 consists of two adjacent triangles with a common edge ei1≤i≤4 such that four edges e₁, e₂, e₃, and e₄ form a matching (see Figure 7D for the four diamonds D₁ − D₄). 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.