On The Matching Forcing And Anti-forcing Numbers Of (3,6)-fullerenes And Hypercubes

The forcing problem of a graph appears in a series of substructures and cor-responding problems such as perfect matchings, dominating sets, coloring and so on. Let M be a perfect matching of a graph G. If S C M and S is not in any other perfect matchings of G, then S is called a forcing set of M. The cardinality of a smallest forcing set is called the forcing number of M. If S(?) C E(G)\M and G-S has a unique perfect matching, then S is called an anti-forcing set of M. The cardinality of a smallest anti-forcing set of M is called the anti-forcing number of M. The smallest value of the forcing numbers of all perfect matchings of G is called the minimum forcing number of G, denoted by f(G). The smallest value of the anti-forcing numbers of all perfect matchings of G is called the anti-forcing number of G, denoted by af(G). The set of the anti-forcing numbers of all perfect matchings of G is called the anti-forcing spectrum of G.(3,6)-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons. In this paper, we mainly considered the forcing number and the anti-forcing number of (3,6)-fullerenes, and the anti-forcing spectrum of Qn. A (3,6)-fullerene graph G has the same connectivity and edge-connectivity 2 or 3. Consequently, we distribute (3,6)-fullerene graph according to its connectivity and study it. For (3,6)-fullerene graph G, we show that f(G)> 2 if the connectivity of G is 3 and G is not isomorphic to K4, otherwise f(G)=1. Then we obtain that af(G)=2 if and only if the connectivity of G is 2 or G is isomorphic to K4, and of(G)≥3 if and only if G has the connectivity 3 and G is not isomorphic to K4. Further, we determine all the (3,6)-fullerenes with anti-forcing number 3. Qn is a n-hypercube, for n≥3, we find a subset of its anti-forcing spectrum, which consists of an arithmetic progression of tolerance n-2. In particular,we proved that the anti-forcing spectrum of Q3 (Q4) exactly is this subset.