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Instead of groups, groupoids, the mathematical generalization of groups in category theory, is the appropriate tool for studying quasicrystals. [51] Using mathematics for construction and analysis of quasicrystal structures is a difficult task. Computer modeling, based on the existing theories of quasicrystals, however, greatly facilitated this ...
The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable. More generally, for a function f assigning a positive integer f(v) to each vertex v, a graph G is f-choosable (or f-list-colorable) if it has a list coloring no matter how one assigns a list of f(v) colors to ...
Several variants of the problem, depending on the particular definitions of nonperiodicity and the specifications of what sets may qualify as tiles and what types of matching rules are permitted, were solved beginning in the 1990s. The strictest version of the problem was solved in 2023, after an initial discovery in 2022.
Its authors have divided Elementary Number Theory, Group Theory and Ramanujan Graphs into four chapters. The first of these provides background in graph theory, including material on the girth of graphs (the length of the shortest cycle), on graph coloring, and on the use of the probabilistic method to prove the existence of graphs for which both the girth and the number of colors needed are ...
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are a well-known example of aperiodic tilings. [1] [2]
A Penrose tiling with rhombi exhibiting fivefold symmetry. A Penrose tiling is an example of an aperiodic tiling.Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches.
A drawing of a graph with 6 vertices and 7 edges.. In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
For instance, α(G) is the independence number of a graph; α′(G) is the matching number of the graph, which equals the independence number of its line graph. Similarly, χ(G) is the chromatic number of a graph; χ ′(G) is the chromatic index of the graph, which equals the chromatic number of its line graph.