{"title":"Coverings by rook domains","authors":"Eugene R. Rodemich","doi":"10.1016/S0021-9800(70)80018-7","DOIUrl":"10.1016/S0021-9800(70)80018-7","url":null,"abstract":"<div><p>It is shown that a 1-dense set in <em>V<sub>n</sub><sup>k</sup></em> must contain at least <em>n<sup>k−1</sup>/(k−1)</em> points. As a corollary, a conjecture of Golomb and Posner on error-distributing codes is proved. It is also shown that a (<em>k</em>−2)-dense set must contain at least <em>n</em><sup>2</sup>/(<em>k</em>−1) points. Equality can be attained if and only if <em>k</em>−1 divides <em>n</em> and there are <em>k</em>−2 orthogonal latin squares of order <em>n</em>/(<em>k</em>−1).</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 117-128"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80018-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88210698","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":"Non-isomorphic solutions of some balanced incomplete block designs. I","authors":"Vasanti N. Bhat, S.S. Shrikhande","doi":"10.1016/S0021-9800(70)80024-2","DOIUrl":"https://doi.org/10.1016/S0021-9800(70)80024-2","url":null,"abstract":"<div><p>In this paper we develop a method for generating non-isomorphic solutions of balanced incomplete block designs belonging to the series of symmetric designs with parameters (4<em>t</em>+3, 2<em>t</em>+1, <em>t</em>) and to the series with parameters (4<em>t</em>+4, 8<em>t</em>+6, 4<em>t</em>+3, 2<em>t</em>+2, 2<em>t</em>+1). We also prove a result about the number of non-isomorphic solutions of these designs as the parameter <em>t</em> tends to infinity.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 174-191"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80024-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137009422","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":"On the existence of regular n-graphs with given girth","authors":"N. Sauer","doi":"10.1016/S0021-9800(70)80021-7","DOIUrl":"10.1016/S0021-9800(70)80021-7","url":null,"abstract":"<div><p>In this paper I construct for each <em>g, l</em>, and <em>m</em>≡0 modulo <em>n</em> a regular <em>n</em>-graph <em>G</em> of degree <em>g</em> and girth <em>l</em> with <em>m≥φ(g, l, n)</em> points, where <em>φ(g, l, n)</em> is a certain function.</p><p>In [1] Erdös constructed such graphs for <em>n</em>=2.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 144-147"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80021-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77837040","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":"Transversals of infinite families with finitely many infinite members","authors":"Jon Folkman","doi":"10.1016/S0021-9800(70)80026-6","DOIUrl":"10.1016/S0021-9800(70)80026-6","url":null,"abstract":"<div><p>The main theorem of this memorandum gives necessary and sufficient conditions for an infinite family of sets with only finitely many infinite members to have a transversal. We also show that the existence of a transversal of an infinite family of sets with only countably many infinite members is equivalent to the existence of a function from the subsets of the index set of the family to the cardinal numbers having certain properties.</p><p>In the introduction we state the most important known results in this area. The final section contains two examples that illustrate the difficulties encountered in attempting to generalize the result we have obtained.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 200-220"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80026-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79968672","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":"Characterizations of derived graphs","authors":"Lowell W. Beineke","doi":"10.1016/S0021-9800(70)80019-9","DOIUrl":"10.1016/S0021-9800(70)80019-9","url":null,"abstract":"<div><p>The derived graph of a graph <em>G</em> has the edges of <em>G</em> as its vertices, with adjacency determined by the adjacency of the edges in <em>G</em>. A new characterization of derived graphs is given in terms of nine excluded subgraphs. A proof of the equivalence of all known characterizations is also given.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 129-135"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80019-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79312179","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":"L'épaisseur des graphes complets K34 et K40","authors":"Jean Mayer","doi":"10.1016/S0021-9800(70)80023-0","DOIUrl":"10.1016/S0021-9800(70)80023-0","url":null,"abstract":"<div><p>Un graphe est défini comme l'union de plusieurs autres si 1°) l'ensemble de ses sommets est l'union des ensembles des sommets des graphes composants, 2°) l'ensemble de ses arêtes est l'union des ensembles des arêtes des graphes composants. On définit alors l'<em>épaisseur</em> d'un graphe <em>G</em> comme le nombre minimum des graphes planaires dont l'union est isomorphe à <em>G</em>.</p><p>Le problème de l'épaisseur des graphes complets a été résolu par F. Harary et L. W. Beineke pour tous les graphes <em>K<sub>n</sub> (n</em> nombre de sommets) où <em>n≠6m</em>+4. L'épaisseur est connue en outre pour <em>n</em>=4, 10, 28. Elle a été récemment déterminée par A. Hobbs pour <em>n</em>=22.</p><p>L'objet du présent article est de déterminer l'épaisseur des graphes <em>K</em><sub>34</sub> et <em>K</em><sub>40</sub>; celle-ci est égale à la limite inférieure déduite par F. Harary et L. W. Beineke de la formule d'Euler. On peut conjecturer très vraisemblablement que cette valeur limite est aussi vérifiée par les graphes <em>K<sub>6m+4</sub> (m</em>≥7)<span><sup>*</sup></span>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 162-173"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80023-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76839325","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":"Graphs, matroids, and geometric lattices","authors":"David Sachs","doi":"10.1016/S0021-9800(70)80025-4","DOIUrl":"10.1016/S0021-9800(70)80025-4","url":null,"abstract":"<div><p>It is shown that two triply connected graphs are isomorphic if their associated geometric lattices are isomorphic. The notion of vertex in a graph is described in terms of irreducible hyperplanes. Finally, necessary and sufficient conditions are given that a lattice be isomorphic to the geometric lattice associated with a graph.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 192-199"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80025-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76179877","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":"Hereditary and recurrent properties in lattices","authors":"James D. Stein Jr.","doi":"10.1016/S0021-9800(70)80016-3","DOIUrl":"10.1016/S0021-9800(70)80016-3","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 103-107"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80016-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74026383","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":"Orthogonal sets of vectors over Zm","authors":"Alan Zame","doi":"10.1016/S0021-9800(70)80020-5","DOIUrl":"10.1016/S0021-9800(70)80020-5","url":null,"abstract":"<div><p>The theory of Hadamard matrices is concerned with finding maximal sets of orthogonal vectors whose components are +1's and −1's. A natural generalization of this problem is to allow entries other than ±1's or to allow entries from some fixed field or ring. In the case of an arbitrary field or ring there may be “orthogonal” sets of “vectors” whose cardinality exceeds the “dimension” of the space. The purpose of this paper is to obtain an explicit formula for the maximum cardinality of such an orthogonal set when the ring involved is <em>Z<sub>m</sub></em>, the integers modulo <em>m</em>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 136-143"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80020-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80752977","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 combinatorial properties of transformations and their connections with the theory of graphs","authors":"J. Dénes","doi":"10.1016/S0021-9800(70)80017-5","DOIUrl":"10.1016/S0021-9800(70)80017-5","url":null,"abstract":"<div><p>Although many results concerning permutations and permutation groups are known, less attention has been paid to transformations and transformation semigroups. It is true that every abstract group is isomorphic to a permutation group, so that with respect to structure there is not difference between abstract groups and permutation groups. Similarly every abstract semigroup is isomorphic to a transformation semigroup; this has led the author to write some papers on the subject [4, 5]. We restrict ourselves to the finite case, and the aim of this paper is to obtain results in this field by a one-to-one correspondence between transformations and directed graphs. The main results of this paper are as follows: (1) Generalization of the Cauchy formula concerning the number of permutations of degree <em>n</em> with prescribed lengths of cycles. (2) Determination of the number of element triples which are generating systems of the symmetric semigroup of degree <em>n</em>, i.e., the semigroup containing every transformation of degree <em>n</em>. (3) Determination of the expected value of the degree of the main permutation of a random transformation of degree <em>n</em>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 108-116"},"PeriodicalIF":0.0,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80017-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74434548","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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