IA Scholar Query: K-coverable polyhex graphs.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 24 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Tilings of Benzels via the Abacus Bijection
https://scholar.archive.org/work/62s35uqpgzdz3ohk7bl6hijio4
Propp recently introduced regions in the hexagonal grid called benzels and stated several enumerative conjectures about the tilings of benzels using two types of prototiles called stones and bones. We resolve two of his conjectures and prove some additional results that he left tacit. In order to solve these problems, we first transfer benzels into the square grid. One of our primary tools, which we combine with several new ideas, is a bijection (rediscovered by Stanton and White and often attributed to them although it is considerably older) between k-ribbon tableaux of certain skew shapes and certain k-tuples of Young tableaux.Colin Defant, Rupert Li, James Propp, Benjamin Youngwork_62s35uqpgzdz3ohk7bl6hijio4Sat, 24 Sep 2022 00:00:00 GMTMaximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs
https://scholar.archive.org/work/6v5mj4rw6fbwbm5soqhrkcz3wa
Let G be a simple graph with 2n vertices and a perfect matching. The forcing number f(G,M) of a perfect matching M of G is the smallest cardinality of a subset of M that is contained in no other perfect matching of G. Among all perfect matchings M of G, the minimum and maximum values of f(G,M) are called the minimum and maximum forcing numbers of G, denoted by f(G) and F(G), respectively. Then f(G)≤ F(G)≤ n-1. Che and Chen (2011) proposed an open problem: how to characterize the graphs G with f(G)=n-1. Later they showed that for a bipartite graph G, f(G)=n-1 if and only if G is a complete bipartite graph K_n,n. In this paper, we completely solve the problem of Che and Chen, and show that f(G)=n-1 if and only if G is a complete multipartite graph or a graph obtained from complete bipartite graph K_n,n by adding arbitrary edges in the same partite set. For all graphs G with F(G)=n-1, we prove that the forcing spectrum of each such graph G forms an integer interval by matching 2-switches and the minimum forcing numbers of all such graphs G form an integer interval from ⌊n/2⌋ to n-1.Qian qian Liu, He ping Zhangwork_6v5mj4rw6fbwbm5soqhrkcz3waWed, 14 Sep 2022 00:00:00 GMTMinimum-Perimeter Lattice Animals and the Constant-Isomer Conjecture
https://scholar.archive.org/work/vdicf5e5pfblrcwredg2swbizm
We consider minimum-perimeter lattice animals, providing a set of conditions which are sufficient for a lattice to have the property that inflating all minimum-perimeter animals of a certain size yields (without repetitions) all minimum-perimeter animals of a new, larger size. We demonstrate this result on the two-dimensional square and hexagonal lattices. In addition, we characterize the sizes of minimum-perimeter animals on these lattices that are not created by inflating members of another set of minimum-perimeter animals.Gill Barequet, Gil Ben-Shacharwork_vdicf5e5pfblrcwredg2swbizmFri, 26 Aug 2022 00:00:00 GMTMinimal-Perimeter Lattice Animals and the Constant-Isomer Conjecture
https://scholar.archive.org/work/fkhlw43v25e5jpw63kxjl5dol4
We consider minimal-perimeter lattice animals, providing a set of conditions which are sufficient for a lattice to have the property that inflating all minimal-perimeter animals of a certain size yields (without repetitions) all minimal-perimeter animals of a new, larger size. We demonstrate this result on the two-dimensional square and hexagonal lattices. In addition, we characterize the sizes of minimal-perimeter animals on these lattices that are not created by inflating members of another set of minimal-perimeter animals.Gill Barequet, Gil Ben-Shacharwork_fkhlw43v25e5jpw63kxjl5dol4Sat, 14 May 2022 00:00:00 GMTA New Algorithm Based on Colouring Arguments for Identifying Impossible Polyomino Tiling Problems
https://scholar.archive.org/work/el2jw74kevdzpns7nosgbdstwy
Checkerboard colouring arguments for proving that a given collection of polyominoes cannot tile a finite target region of the plane are well-known and typically applied on a case-by-case basis. In this article, we give a systematic mathematical treatment of such colouring arguments, based on the concept of a parity violation, which arises from the mismatch between the colouring of the tiles and the colouring of the target region. Identifying parity violations is a combinatorial problem related to the subset sum problem. We convert the combinatorial problem into linear Diophantine equations and give necessary and sufficient conditions for a parity violation. The linear Diophantine equation approach leads to an algorithm implemented in MATLAB for finding all possible parity violations of large tiling problems, and is the main contribution of this article. Numerical examples illustrate the effectiveness of our algorithm. The collection of MATLAB programs, POLYOMINO_PARITY (v2.0.0) is freely available for download.Marcus R. Garvie, John Burkardtwork_el2jw74kevdzpns7nosgbdstwyThu, 17 Feb 2022 00:00:00 GMTMatching forcing polynomial of generalized Petersen graph GP(n, 2)
https://scholar.archive.org/work/7bfudyvqejal7gfyexytlgm46q
Harary et al. and Klein and Randic proposed the forcing number of a perfect matching in mathematics and chemistry, respectively. In detail, the forcing number of a perfect matching M of a graph G is the smallest cardinality of subsets of M that are contained in no other perfect matchings of G. The author and cooperators defined the forcing polynomial of G as the count polynomial for perfect matchings with the same forcing number of G, from which the average forcing number, forcing spectrum, and the maximum and minimum forcing numbers of G can be obtained. Up to now, a few papers have been considered on matching forcing problem of non-plane non-bipartite graphs. In this paper, we investigate the forcing polynomials of generalized Petersen graphs GP(n, 2) for n = 5, 6, . . . , 15, which is a typical class of non-plane non-bipartite graph.Shuang Zhaowork_7bfudyvqejal7gfyexytlgm46qFri, 08 Oct 2021 00:00:00 GMT