Pub Date : 1970-07-01DOI: 10.1016/S0021-9800(70)80055-2
Solomon W. Golomb
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.
{"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}
Pub Date : 1970-07-01DOI: 10.1016/S0021-9800(70)80060-6
W.D. Wallis
A quasi-symmetric balanced incomplete block design with parameters (4y, 8y−2, 4y−1, 2y, 2y−1) exists if and only if there is an Hadamard matrix of order 4y.
{"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}
Pub Date : 1970-06-01DOI: 10.1016/S0021-9800(70)80033-3
László Lovász
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].
{"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}
Pub Date : 1970-06-01DOI: 10.1016/S0021-9800(70)80029-1
William G. Bridges Jr. , Earl S. Kramer
Let S1, …, Sn, n>1, be subsets of an n-set S where |Si|>λ≥1 and |Si∩Sj|=λ for i≠j. 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 S occurs either r1 or r2 times among the sets Si, …, Sn and r1+r2=n+1. The 1-designs are completely known and so is the unique 2-design. The present paper establishes that there are exactly three 3-designs.
{"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}
Pub Date : 1970-06-01DOI: 10.1016/S0021-9800(70)80037-0
Don R. Lick
{"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}
Pub Date : 1970-06-01DOI: 10.1016/S0021-9800(70)80032-1
Donald E. Knuth
{"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}
Pub Date : 1970-06-01DOI: 10.1016/S0021-9800(70)80039-4
W.D. Wallis
We show that there are an infinitude of values (v, k, λ) for which there are a pair of non-isomorphic (v, k, λ)-graphs.
我们证明存在无穷个值(v, k， λ)，对于这些值存在一对非同构的(v, k， λ)-图。
{"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}
Pub Date : 1970-06-01DOI: 10.1016/S0021-9800(70)80040-0
{"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}
Pub Date : 1970-06-01DOI: 10.1016/S0021-9800(70)80030-8
W.G. Bridges
A λ-design as introduced by Ryser [3] is a (0, 1)-square matrix with constant column inner products but not 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 H-designs. This paper does three things: (1) generalizes Ryser's H-design construction to an arbitrary (ν, k, λ)-configuration, (2) establishes some additional general properties of λ-designs, and (3) determines all 4-designs.
{"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}
Pub Date : 1970-06-01DOI: 10.1016/S0021-9800(70)80036-9
Peter C. Fishburn
{"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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