论文标题
二次圆柱体和圆环的翻转下的多米诺骨牌的组成部分
Components of domino tilings under flips in quadriculated cylinder and torus
论文作者
论文摘要
在一个由单位广场组成的区域$ r $中，多米诺骨牌是两个相邻广场的结合，而（多米诺骨牌）的瓷砖是多米诺骨牌的集合，其内部位于该区域。 Flip Graph $ \ MATHCAL {t}（r）$是在$ r $的所有瓷砖的集合中定义的，这样，如果我们通过翻转将一个块更改为另一个瓷砖（$ 90^{\ circ} $旋转一对侧面多米诺骨牌）。众所周知，当$ r $仅连接时，$ \ natercal {t}（r）$连接。通过使用图理论方法，我们表明$ 200M \ times（2n+1）的翻转图仍连接起来，但是$ 200M \ times（2n+1）的翻转图已断开，并由两个同构成组成。对于平铺$ t $，我们将一个整数$ f（t）$（迫使数字）关联为$ t $中最小的多米诺骨牌数量，其中包含在没有其他瓷砖中。作为一个应用程序，我们获得了所有瓷砖的强迫数量为$ 200M \ times（2n+1）$二次圆柱体和圆环形成一个整数间隔，其最大值为$（n+1）m $。
In a region $R$ consisting of unit squares, a domino is the union of two adjacent squares and a (domino) tiling is a collection of dominoes with disjoint interior whose union is the region. The flip graph $\mathcal{T}(R)$ is defined on the set of all tilings of $R$ such that two tilings are adjacent if we change one to another by a flip (a $90^{\circ}$ rotation of a pair of side-by-side dominoes). It is well-known that $\mathcal{T}(R)$ is connected when $R$ is simply connected. By using graph theoretical approach, we show that the flip graph of $2m\times(2n+1)$ quadriculated cylinder is still connected, but the flip graph of $2m\times(2n+1)$ quadriculated torus is disconnected and consists of exactly two isomorphic components. For a tiling $t$, we associate an integer $f(t)$, forcing number, as the minimum number of dominoes in $t$ that is contained in no other tilings. As an application, we obtain that the forcing numbers of all tilings in $2m\times (2n+1)$ quadriculated cylinder and torus form respectively an integer interval whose maximum value is $(n+1)m$.
