Some novel minimax results for perfect matchings of hexagonal systems

Abstract

The anti-forcing number of a perfect matching M of a graph G is the minimum number of edges of G whose deletion results in a subgraph with a unique perfect matching M , denoted by a f ( G , M ) . When G is a plane bipartite graph, Lei et al. established a minimax result: For any perfect matching M of G , a f ( G , M ) equals the maximum number of M -alternating cycles of G where any two either are disjoint or intersect only at edges in M ; For a hexagonal system, the maximum anti-forcing number equals the Fries number. In this paper we show that for every perfect matching M of a hexagonal system H with the maximum anti-forcing number or minus one, a f ( H , M ) equals the number of M -alternating hexagons of H . Further we show that a hexagonal system H has no Triphenylene as a nice subgraph if and only if a f ( H , M ) always equals the number of M -alternating hexagons of H for every perfect matching M of H .