A Maximum Resonant Set of Polyomino Graphs

A polyomino graph $H$ is a connected finite subgraph of the infinite plane grid such that each finite face is surrounded by a regular square of side length one and each edge belongs to at least one square. In this paper, we show that if $K$ is a maximum resonant set of $H$, then $H-K$ has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to its Clar number. Based on this result, we have that the maximum forcing number of a polyomino graph can be computed in polynomial time. We also show that if $K$ is a maximal alternating set of $H$, then $H-K$ has a unique perfect matching.Comment: 13 pages, 6 figure

Components of domino tilings under flips in quadriculated cylinder and torus

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 $(n+1)m$

Geometric and algebraic properties of polyomino tilings

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.Includes bibliographical references (p. 165-167).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.In this thesis we study tilings of regions on the square grid by polyominoes. A polyomino is any connected shape formed from a union of grid cells, and a tiling of a region is a collection of polyominoes lying in the region such that each square is covered exactly once. In particular, we focus on two main themes: local connectivity and tile invariants. Given a set of tiles T and a finite set L of local replacement moves, we say that a region [Delta] has local connectivity with respect to T and L if it is possible to convert any tiling of [Delta] into any other by means of these moves. If R is a set of regions (such as the set of all simply connected regions), then we say there is a local move property for T and R if there exists a finite set of moves L such that every r in R has local connectivity with respect to T and L. We use height function techniques to prove local move properties for several new tile sets. In addition, we provide explicit counterexamples to show the absence of a local move property for a number of tile sets where local move properties were conjectured to hold. We also provide several new results concerning tile invariants. If we let ai(t) denote the number of occurrences of the tile ti in a tiling t of a region [Delta], then a tile invariant is a linear combination of the ai's whose value depends only on t and not on r.(cont.) We modify the boundary-word technique of Conway and Lagarias to prove tile invariants for several new sets of tiles and provide specific examples to show that the invariants we obtain are the best possible. In addition, we prove some new enumerative results, relating certain tiling problems to Baxter permutations, the Tutte polynomial, and alternating-sign matrices.by Michael Robert Korn.Ph.D

Realizations of multiassociahedra via bipartite rigidity

Let $Ass_k(n)$ denote the simplicial complex of $(k+1)$-crossing-free subsets of edges in $\binom{n}{2}$. Here $k,n\in \mathbb{N}$ and $n\ge 2k+1$. It is conjectured that this simplicial complex is polytopal (Jonsson 2005). However, despite several recent advances, this is still an open problem. In this paper we attack this problem using as a vector configuration the rows of a rigidity matrix, namely, hyperconnectivity restricted to bipartite graphs. We see that in this way $Ass_k(n)$ can be realized as a polytope for $k=2$ and $n\le 10$, and as a fan for $k=2$ and $n\le 13$, and for $k=3$ and $n\le 11$. However, we also prove that the cases with $k\ge 3$ and $n\ge \max\{12,2k+4\}$ are not realizable in this way. We also give an algebraic interpretation of the rigidity matroid, relating it to a projection of determinantal varieties with implications in matrix completion, and prove the presence of a fan isomorphic to $Ass_{k-1}(n-2)$ in the tropicalization of that variety.Comment: 30 pages, 2 figures. arXiv admin note: text overlap with arXiv:2212.1426

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Exact sampling with Markov chains

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (p. 79-83).by David Bruce Wilson.Ph.D