{"title":"Tiling with sets of polyominoes","authors":"Solomon W. Golomb","doi":"10.1016/S0021-9800(70)80055-2","DOIUrl":"10.1016/S0021-9800(70)80055-2","url":null,"abstract":"<div><p>The definitions and lattice hierarchy previously established for tiling regions with individual polyominoes are extended to finite sets of polyominoes. The problem of tiling the infinite plane with replicas of a finite set of polyominoes is proved to be logically equivalent to Wang's “domino problem,” which is known to be algorithmically undecidable. Several different ways of extending the notion of rep-tility from single polyominoes to sets of polyominoes are discussed. Some related results of Ikeno regarding tiling with polyiamonds (shapes composed of equilateral triangles) are mentioned.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 60-71"},"PeriodicalIF":0.0,"publicationDate":"1970-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80055-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73339351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on quasi-symmetric designs","authors":"W.D. Wallis","doi":"10.1016/S0021-9800(70)80060-6","DOIUrl":"10.1016/S0021-9800(70)80060-6","url":null,"abstract":"<div><p>A quasi-symmetric balanced incomplete block design with parameters (4<em>y</em>, 8<em>y</em>−2, 4<em>y</em>−1, 2<em>y</em>, 2<em>y</em>−1) exists if and only if there is an Hadamard matrix of order 4<em>y</em>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 100-101"},"PeriodicalIF":0.0,"publicationDate":"1970-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80060-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87017344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Subgraphs with prescribed valencies","authors":"László Lovász","doi":"10.1016/S0021-9800(70)80033-3","DOIUrl":"10.1016/S0021-9800(70)80033-3","url":null,"abstract":"<div><p>In this paper a generalization of the factor problem for finite undirected graphs is detailed. We prescribe certain inequalities for the valencies of a subgraph. We deduce formulas for the minimum “deviation” of this prescription and characterize the “optimally approaching” subgraphs. These results include the conditions of Tutte and Ore for the existence of a factor and the characterization of maximal independent edge-systems given in [3] and [11].</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 4","pages":"Pages 391-416"},"PeriodicalIF":0.0,"publicationDate":"1970-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80033-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89589960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The determination of all λ-designs with λ=3","authors":"William G. Bridges Jr. , Earl S. Kramer","doi":"10.1016/S0021-9800(70)80029-1","DOIUrl":"10.1016/S0021-9800(70)80029-1","url":null,"abstract":"<div><p>Let <em>S</em><sub>1</sub>, …, <em>S<sub>n</sub></em>, <em>n</em>>1, be subsets of an <em>n</em>-set <em>S</em> where |<em>S<sub>i</sub></em>|>λ≥1 and |<em>S<sub>i</sub></em>∩<em>S<sub>j</sub></em>|=λ for <em>i≠j</em>. Then our configuration is either a symmetric block design, with possible degeneracies, or what Ryser [3] has called a λ-design. A λ-design has the remarkable property, established by Ryser [3], that each element of <em>S</em> occurs either <em>r</em><sub>1</sub> or <em>r</em><sub>2</sub> times among the sets <em>S<sub>i</sub></em>, …, <em>S<sub>n</sub></em> and <em>r<sub>1</sub>+r<sub>2</sub>=n+1</em>. The 1-designs are completely known and so is the unique 2-design. The present paper establishes that there are exactly three 3-designs.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 4","pages":"Pages 343-349"},"PeriodicalIF":0.0,"publicationDate":"1970-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80029-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73010058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A sufficient condition for hamiltonian connectedness","authors":"Don R. Lick","doi":"10.1016/S0021-9800(70)80037-0","DOIUrl":"10.1016/S0021-9800(70)80037-0","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 4","pages":"Pages 444-445"},"PeriodicalIF":0.0,"publicationDate":"1970-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80037-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74256066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Notes on central groupoids","authors":"Donald E. Knuth","doi":"10.1016/S0021-9800(70)80032-1","DOIUrl":"10.1016/S0021-9800(70)80032-1","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 4","pages":"Pages 376-390"},"PeriodicalIF":0.0,"publicationDate":"1970-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80032-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72810567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some non-isomorphic graphs","authors":"W.D. Wallis","doi":"10.1016/S0021-9800(70)80039-4","DOIUrl":"10.1016/S0021-9800(70)80039-4","url":null,"abstract":"<div><p>We show that there are an infinitude of values (<em>v, k, λ</em>) for which there are a pair of non-isomorphic (<em>v, k, λ</em>)-graphs.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 4","pages":"Pages 448-449"},"PeriodicalIF":0.0,"publicationDate":"1970-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80039-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83174514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Author index of volume 8","authors":"","doi":"10.1016/S0021-9800(70)80040-0","DOIUrl":"https://doi.org/10.1016/S0021-9800(70)80040-0","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 4","pages":"Pages 450-451"},"PeriodicalIF":0.0,"publicationDate":"1970-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80040-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137420574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some results on λ-designs","authors":"W.G. Bridges","doi":"10.1016/S0021-9800(70)80030-8","DOIUrl":"10.1016/S0021-9800(70)80030-8","url":null,"abstract":"<div><p>A λ-design as introduced by Ryser [3] is a (0, 1)-square matrix with constant column inner products but <em>not</em> all column sums equal. Ryser has shown such a matrix to have two row sums and he constructs an infinite family of λ-designs called <em>H</em>-designs. This paper does three things: (1) generalizes Ryser's <em>H</em>-design construction to an arbitrary (ν, <em>k</em>, λ)-configuration, (2) establishes some additional general properties of λ-designs, and (3) determines all 4-designs.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 4","pages":"Pages 350-360"},"PeriodicalIF":0.0,"publicationDate":"1970-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80030-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81009878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An interval graph is not a comparability graph","authors":"Peter C. Fishburn","doi":"10.1016/S0021-9800(70)80036-9","DOIUrl":"10.1016/S0021-9800(70)80036-9","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 4","pages":"Pages 442-443"},"PeriodicalIF":0.0,"publicationDate":"1970-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80036-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91090117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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