The anti-forcing spectra of [formula omitted]-fullerenes

Abstract

For an edge subset S of connected graph G, if G−S has only one perfect matching M, then S is called an anti-forcing set of M. The number of edges in a smallest anti-forcing set of M is the anti-forcing number of M, generally indicated by the symbol af(G,M). For a graph G, its anti-forcing spectrum is defined as the integer set Specaf(G):={af(G,M):M is a perfect matching of G}. In this paper, we show that for a (4,6)-fullerene graph Tn with cyclic edge-connectivity 3, Specaf(Tn)=[n+3,2n+4]. Moreover, we show that for any perfect matching M of a (4,6)-fullerene graph G, a minimum anti-forcing set S of M and each M-alternating facial boundary share exactly one edge. Applying this conclusion, we prove that the minimum anti-forcing number of a lantern structure (4,6)-fullerene of order n is ⌊n8⌋+2, and Specaf(Bn)=[⌊n8⌋+2,⌊n3⌋+2].