Related papers: Saturated Domino Coverings
It is broadly known that any parallelepiped tiles space by translating copies of itself along its edges. In earlier work relating to higher-dimensional sandpile groups, the second author discovered a novel construction which fragments the…
In this paper, we consider domino tilings of regions of the form $\mathcal{D} \times [0,n]$, where $\mathcal{D}$ is a simply connected planar region and $n \in \mathbb{N}$. It turns out that, in nontrivial examples, the set of such tilings…
A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or…
We consider asymtotics of a domino tiling model on a class of domains which we call rectangular Aztec diamonds. We prove the Law of Large Numbers for the corresponding height functions and provide explicit formulas for the limit. For a…
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…
Given a dominating set, how much smaller a dominating set can we find through elementary operations? Here, we proceed by iterative vertex addition and removal while maintaining the property that the set forms a dominating set of bounded…
Domination is the fastest-growing field within graph theory with a profound diversity and impact in real-world applications, such as the recent breakthrough approach that identifies optimized subsets of proteins enriched with cancer-related…
{\em Partial domination problem} is a generalization of the {\em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a…
We prove several results of the following type: given finite dimensional normed space V there exists another space X with log (dim X) = O(log (dim V)) and such that every subspace (or quotient) of X, whose dimension is not "too small,"…
You find a stain on the wall and decide to cover it with non-overlapping stickers of a single identical shape (rotation and reflection are allowed). Is it possible to find a sticker shape that fails to cover the stain? In this paper, we…
The Queen's Domination problem, studied for over 160 years, poses the following question: What is the least number of queens that can be arranged on a $m \times n$ chessboard so that they either attack or occupy every cell? We propose a…
We study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it…
In a recent paper, Cho and Kim proved that in subcubic graphs, the independent domination number is at most three times the packing number. They subsequently posed the question of characterizing subcubic graphs that achieve this bound. In…
A dominating set $S$ is an Isolate Dominating Set (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate…
This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which…
Inspired by Propp's intruded Aztec diamond regions, we consider halved hexagons in which two aligned arrays of triangular holes have been removed from their boundaries. Unlike the intruded Aztec diamonds (whose numbers of domino tilings…
In a directed graph $D$, a vertex subset $S\subseteq V$ is a total dominating set if every vertex of $D$ has an in-neighbor from $S$. A total dominating set exists if and only if every vertex has at least one in-neighbor. We call the…
There is a natural generalization of domino tilings to tilings of a polygon by hexagons, or, dually, configurations of oriented curves that meet in triples. We show exactly when two such tilings can be connected by a series of moves…
A famous result of D. Walkup is that an $m\times n$ rectangle may be tiled by T-tetrominos if and only if both $m$ and $n$ are multiples of 4. The "if" portion may be proved by tiling a $4\times 4$ block, and then copying that block to fill…
We present some elementary but foundational results concerning diamond-colored modular and distributive lattices and connect these structures to certain one-player combinatorial "move-minimizing games," in particular, a so-called "domino…