![]() We establish upper bounds on the tessellation cover number given by the minimum between the chromatic index of the graph and the chromatic number of its clique graph and we show graph classes for which these bounds are tight. This problem is motivated by its applications to quantum walk models, in especial, the evolution operator of the staggered model is obtained from a graph tessellation cover. The $t$-tessellability problem aims to decide whether there is a tessellation cover of the graph with $t$ tessellations. A tessellation cover of a graph is a set of tessellations that covers all of its edges. Abreu and 7 other authors Download PDF Abstract:A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. It doesn't include keeping the trapezoids on the same side and adding mirror symmetries or seeing if I can use different triangle twists in the same tessellation.Download a PDF of the paper titled The graph tessellation cover number: extremal bounds, efficient algorithms and hardness, by A. It also doesn't include longer-range repeats, adding spacing between particular repeats, or adding spacing in particular directions. Now, I'm only seeing three obvious variations on Woven Strips, but that doesn't include changing spacing or going to more open triangle twists. Nearly every week I figure out some new system of tessellations - usually with at least a dozen possible variations. In fact, there are thousands of tessellations waiting to be discovered. Whenever I'm talking to people unfamiliar with tessellations, they have an intuition that the tessellations have all been discovered - that there's nothing new left to find. The "hexagon" in this case is fixed as the isoarea pair of trapezoid twists - all that remains is a choice of closed or open triangle twist, on the same or opposite side as the closest twist. The twist is that the two trapezoids in the pair are on opposite sides of the paper, and so you get the same number of each kind of twist on each side.Īs far as I can tell, this tiling will have the same kinds of variations as the simple hexagons and triangles tiling. This trapezoids and triangles tiling is very similar to the common tiling of hexagons and triangles with 6-fold rotational symmetry.Įach pair of trapezoids fits into one hexagon of the more common tiling and so, just like that tiling, this tessellation has six triangle twists around each pair of trapezoids. In the vertical orientation it looks like a chain of buckets, raising and lowering some cargo. In this particular orientation it also looks like thrust faults and rolling waves of tsunamis. ![]() It's as if the trapezoid twists are pushing the slanted bars into place, like a rotating portion of a machine. We see these shifting horizontal bars in the pattern, crossed by slanted bars. While I have done the math to align this kind of pattern on this kind of grid, I haven't tested it yet in practice - which is why this piece isn't aligned with the repeats. This alignment question comes into sharp relief with patterns like Woven Strips where the pattern repeats in stripes that are very clearly not oriented in alignment with the edges of the paper! That's where rotated grids come into play. What if we want to align our grid to the pattern repeats? So, what if we want to align our grid to something else? We don't say "square grid aligned with the edges of the paper" - it's just "a square grid".īut really a grid is an abstract concept and we have to make decisions about how we're going to apply that concept to the paper in front of us. This fact is invisible in the way we talk about grids too. ![]() When we start folding tessellations we're presented with a couple grid options - all of which are aligned with the edge of the paper in some way. ![]()
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