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 $$ . In this paper, we show that $$ ; 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 Q is composed of $$ concentric layers of quadrangles, capped on each end by a cap formed by four triangles which share a common vertex (see Figure 2 for the graph Q ); and the other is the graph $$ , which contains four diamonds D ₁, D ₂, D ₃, and D ₄, where each diamond $$ consists of two adjacent triangles with a common edge $$ 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 $$ , then $$ ; moreover, if $$ , we give the condition to judge that the anti-Kekulé number of graph G is 4 or 5.