Tight upper bound on the maximum anti-forcing numbers of graphs

Let $G$ be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of $G$ is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of $G$ and investigate the extremal graphs. If $G$ has a perfect matching $M$ whose anti-forcing number attains this upper bound, then we say $G$ is an extremal graph and $M$ is a nice perfect matching. We obtain an equivalent condition for the nice perfect matchings of $G$ and establish a one-to-one correspondence between the nice perfect matchings and the edge-involutions of $G$, which are the automorphisms $\alpha$ of order two such that $v$ and $\alpha(v)$ are adjacent for every vertex $v$. We demonstrate that all extremal graphs can be constructed from $K_2$ by implementing two expansion operations, and $G$ is extremal if and only if one factor in a Cartesian decomposition of $G$ is extremal. As examples, we have that all perfect matchings of the complete graph $K_{2n}$ and the complete bipartite graph $K_{n, n}$ are nice. Also we show that the hypercube $Q_n$, the folded hypercube $FQ_n$ ($n\geq4$) and the enhanced hypercube $Q_{n, k}$ ($0\leq k\leq n-4$) have exactly $n$, $n+1$ and $n+1$ nice perfect matchings respectively.Comment: 15 pages, 7 figure

Relations between global forcing number and maximum anti-forcing number of a graph

The global forcing number of a graph G is the minimal cardinality of an edge subset discriminating all perfect matchings of G, denoted by gf(G). For any perfect matching M of G, the minimal cardinality of an edge subset S in E(G)-M such that G-S has a unique perfect matching is called the anti-forcing number of M,denoted by af(G, M). The maximum anti-forcing number of G among all perfect matchings is denoted by Af(G). It is known that the maximum anti-forcing number of a hexagonal system equals the famous Fries number. We are interested in some comparisons between the global forcing number and the maximum anti-forcing number of a graph. For a bipartite graph G, we show that gf(G)is larger than or equal to Af(G). Next we mainly extend such result to non-bipartite graphs, which is the set of all graphs with a perfect matching which contain no two disjoint odd cycles such that their deletion results in a subgraph with a perfect matching. For any such graph G, we also have gf(G) is larger than or equal to Af(G) by revealing further property of non-bipartite graphs with a unique perfect matching. As a consequence, this relation also holds for the graphs whose perfect matching polytopes consist of non-negative 1-regular vectors. In particular, for a brick G, de Carvalho, Lucchesi and Murty [4] showed that G satisfying the above condition if and only if G is solid, and if and only if its perfect matching polytope consists of non-negative 1-regular vectors. Finally, we obtain tight upper and lower bounds on gf(G)-Af(G). For a connected bipartite graph G with 2n vertices, we have that 0 \leq gf(G)-Af(G) \leq 1/2 (n-1)(n-2); For non-bipartite case, -1/2 (n^2-n-2) \leq gf(G)-Af(G) \leq (n-1)(n-2).Comment: 19 pages, 11 figure

Fractional forcing number of graphs

The notion of forcing sets for perfect matchings was introduced by Harary, Klein, and \v{Z}ivkovi\'{c}. The application of this problem in chemistry, as well as its interesting theoretical aspects, made this subject very active. In this work, we introduce the notion of the forcing function of fractional perfect matchings which is continuous analogous to forcing sets defined over the perfect matching polytope of graphs. We show that our defined object is a continuous and concave function extension of the integral forcing set. Then, we use our results about this extension to conclude new bounds and results about the integral case of forcing sets for the family of edge and vertex-transitive graphs and in particular, hypercube graphs