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

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 [Formula: see text] . In this paper, we show that [Formula: see text] ; 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 [Formula: see text] consisting of the tubular graph [Formula: see text] , where Q ( n ) is composed of [Formula: see text] 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 [Formula: see text] , which contains four diamonds D (1), D (2), D (3), and D (4), where each diamond [Formula: see text] consists of two adjacent triangles with a common edge [Formula: see text] 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 [Formula: see text] , then [Formula: see text] ; moreover, if [Formula: see text] , we give the condition to judge that the anti-Kekulé number of graph G is 4 or 5.