Constructing The Fullerene Graphs With The Minimum Anti-forcing Number 5

The anti-forcing number of a connected graph G is the smallest number of edges such that the remaining graph by removing these edges has a unique perfect matching.Fullerene graph Fn is a cubic 3-connected plane graph with only 12 pentagonal faces and hexagonal faces.By Euler’s formula,every fullerene graph has exactly twelve pentagonal faces.Grunbaum and Motzkin[12]showed that a fullerene graph with n vertices exists for n=20 and for all even n>24.Yang et al proved that the anti-forcing numbers of every fullerene graphs are at least four and constructed the fullerene graphs with the anti-forcing number 4,and implied that there exists a fullerene Fn such that af(Fn)=4 for any even n≥ 20(n≠ 22,26),and fullerene graphs F26 have the anti-forcing number 5.In this paper,we construct fullerene graphs with the minimum anti-forcing numbers 5 by making operations O1-O5 on the boundary of every initial seed graph to get a larger generalized patch.We find that for a fullerene graph F with the minimum anti-forcing numbers 5,its minimum anti-forcing set E’ is not a matching.Let F" be the remaining graph after deleting both end vertices of all the pendent edges from F-E’,and F’ the subgraph induced by all deleted vertices of F,and X the set of edges of F from F’ to F".We get that |X|=6,8 or 10 and the corresponding F" when F"≠(?).Furthermore,we get some certain fullerene graphs and three classes of fullerene graphs with the anti-forcing number 5.For the three classes of fullerene graphs,when the vertices n>30,the corresponding fullerene graphs have the anti-forcing number 5,except the fullerene graph F34 of the first class of fullerene graphs.