Anti-Kekulé number of the {(3, 4), 4}-fullerene.

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.