Anti-Kekulé number of the (3, 4), 4-fullerene* - Yang, Rui; Jia, Huimin (Frontiers in Chemistry, 2023)
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-Kekule set, if G-S is a connected subgraph without perfect matchings. The anti-Kekule number of G is the smallest cardinality of anti-Kekule sets and is denoted by a( G). In this paper, we show that a( G) is 4 or 5, at the same time, we determine that the (3,4),4-fullerene graph with the anti-Kekule number 4 consists two kinds of graphs, one of which is the graph H_1 consisting of the tubular graph Q_n (n>=0), where Q_n is composed of n concentric layers of quadrangles, capped on each end by a cap formed by four triangles which share a common vertex (see Fig.2 the graph Q_n), the other of which is the graph H_2, which is containing four diamonds D_1, D_2, D_3 and D_4, where each diamond D_i is consisting of two adjacent triangles with a common edge e_i such that four edges e_1, e_2, e_3, e_4 forms a matching (see Fig.7.(d) the four diamonds D_1-D_4). As a consequence, we proof that if G belongs to H_1, then a(G) = 4, moreover, if G belongs to H_2, we give the condition to judge that the anti-Kekule number of graph G is 4 or 5.