Hamilton paths in Z-transformation graphs of perfect matchings of hexagonal systems

AbstractLet H be a hexagonal system. The Z-transformation graph Z(H) is the graph where the vertices are the perfect matchings of H and where two perfect matchings are joined by an edge provided their symmetric difference is a hexagon of H (Z. Fu-ji et al., 1988). In this paper we prove that Z(H) has a Hamilton path if H is a catacondensed hexagonal system

Decomposition theorem on matchable distributive lattices

A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph $G$ is elementary, then ${\mathbf M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ is an MDL. Two types of MDLs are presented: $J(\mathbf{m}\times \mathbf{n})$ and $J(\mathbf{T})$, where $\mathbf{m}\times \mathbf{n}$ denotes the cartesian product between $m$-element chain and $n$-element chain, and $\mathbf{T}$ is a poset implied by any orientation of a tree.Comment: 19 pages, 7 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

Total forcing number of the triangular grid

Let $T$ be a square triangular grid with $n$ rows and columns of vertices and $n$ an even number. A set of edges $Esubset E(T)$ completely determines perfect matchings on $T$ if there are no two different matchings on $T$ coinciding on $E.$ We establish the upper and the lower bound for the smallest value of $left| Eright| ,$ i.e. we show that begin{equation*} frac{5}{4}n^{2}-frac{21}{2}n+frac{41}{4}leq left| Eright| leq frac{5}{4}n^{2}+n-2 end{equation*}% and show that $left| Eright| /left| E(T)right|$ tends to $5/12$ when $n$ tends to infinity

The anti-forcing number of double hexagonal chains

Abstract The anti-forcing number of a graph is the smallest number of edges that have to be removed so that the remaining graph contains only one perfect matching. In this paper, the anti-forcing number of double hexagonal chains is determined and the extremal graphs are characterized