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

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 Qn 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 Qn); 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.