{"title":"The crossing number of K5,n","authors":"Daniel J. Kleitman","doi":"10.1016/S0021-9800(70)80087-4","DOIUrl":"10.1016/S0021-9800(70)80087-4","url":null,"abstract":"<div><p>Several arguments are presented which provide restrictions on the possible number of crossings in drawings of bipartite graphs. In particular it is shown that <em>cr(K<sub>5,n</sub>)</em>=4[1/2<em>n</em>][1/2(<em>n</em>−1)] and <em>cr(K<sub>6,n</sub>)</em>=6[1/2<em>n</em>][1/2(<em>n</em>−1)].</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 315-323"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80087-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79235649","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":"Correspondences between plane trees and binary sequences","authors":"David A. Klarner","doi":"10.1016/S0021-9800(70)80093-X","DOIUrl":"10.1016/S0021-9800(70)80093-X","url":null,"abstract":"<div><p>The subject of each of the five sections of this paper is the planted plane trees discussed by Harary, Prins, and Tutte [7]. A description of the content of the present work is given in Section 1. Section 2 is devoted to a definition of plane trees in terms of finite sets and relations defined on them—we hope this definition will replace the topological concepts introduced in [7]. A one-to-one correspondence between the classes of isomorphic planted plane trees with <em>n</em>+2 vertices and the classes of isomorphic 3-valent planted plane trees with 2<em>n</em>+2 vertices is given in Section 3. Sections 4 and 5 deal with enumeration problems.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 401-411"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80093-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82953586","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 nonreconstructable tournaments","authors":"L.W. Beineke , E.T. Parker","doi":"10.1016/S0021-9800(70)80088-6","DOIUrl":"10.1016/S0021-9800(70)80088-6","url":null,"abstract":"<div><p>Pairs of non-isomorphic strong tournaments of orders 5 and 6 are given for which the subtournaments of orders 4 and 5, respectively, are pairwise isomorphic. Herefore, only pairs of orders 3 and 4 were known.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 324-326"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80088-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75122611","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 rank inequality for finite geometric lattices","authors":"Curtis Greene","doi":"10.1016/S0021-9800(70)80090-4","DOIUrl":"10.1016/S0021-9800(70)80090-4","url":null,"abstract":"<div><p>Let <em>L</em> be a finite geometric lattice of dimension <em>n</em>, and let <em>w(k)</em> denote the number of elements in <em>L</em> of rank <em>k</em>. Two theorems about the numbers <em>w(k)</em> are proved: first, <em>w(k)</em>≥<em>w</em>(1) for <em>k</em>=2,3,…,<em>n</em>−1. Second, <em>w(k)</em>=<em>w</em>(1) if and only if <em>k</em>=<em>n</em>−1 and <em>L</em> is modular. Several corollaries concerning the “matching” of points and dual points are derived from these results.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 357-364"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80090-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73432231","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 9","authors":"","doi":"10.1016/S0021-9800(70)80098-9","DOIUrl":"https://doi.org/10.1016/S0021-9800(70)80098-9","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Page 428"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80098-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137003022","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 number of classes of isomorphic graded partially ordered sets","authors":"David A. Klarner","doi":"10.1016/S0021-9800(70)80094-1","DOIUrl":"10.1016/S0021-9800(70)80094-1","url":null,"abstract":"<div><p>This note is a continuation of [2]; we describe here how to enumerate classes of isomorphic graded posets defined on a finite set. Perhaps the most interesting aspect of the results presented here is that the enumeration of these complex structures can be carried out in the algebra of formal power series.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 412-419"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80094-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77802171","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":"Symmetric embeddings of graphs","authors":"D.F. Robinson","doi":"10.1016/S0021-9800(70)80092-8","DOIUrl":"10.1016/S0021-9800(70)80092-8","url":null,"abstract":"<div><p>Let <em>G</em> be a graph, <em>G</em>′ an embedding of <em>G</em> as a straight 1-complex in <em>R</em><sup>n</sup>, the real coordinate space of dimension <em>n</em>; let Φ be a group of transformations mapping <em>R<sup>n</sup></em> to itself. If for every automorphism α of <em>G</em> we can find a member of Φ mapping <em>G</em>′ onto itself in such a way that it induces α in <em>G</em>′, we say that <em>G</em>′ is a Φ-symmetric embedding of <em>G</em>. In particular this paper discusses conditions for the existence of such an embedding when Φ is the group of autohomeomorphisms of <em>R<sup>n</sup></em> or the group of invertible linear transformations in <em>R<sup>n</sup></em>, and the graph is the complete graph <em>K<sub>m</sub></em>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 377-400"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80092-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82589030","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":"Enumeration of non-separable graphs","authors":"Robert W. Robinson","doi":"10.1016/S0021-9800(70)80089-8","DOIUrl":"10.1016/S0021-9800(70)80089-8","url":null,"abstract":"<div><p>Non-separable graphs are enumerated, and also graphs without end-points. The basic enumeration tool is sums of cycle indices of automorphism groups.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 327-356"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80089-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86669100","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":"Regular d-valent graphs of girth 6 and 2(d2−d+1) vertices","authors":"Judith Q. Longyear","doi":"10.1016/S0021-9800(70)80095-3","DOIUrl":"10.1016/S0021-9800(70)80095-3","url":null,"abstract":"<div><p>For each <em>d</em> such that <em>d</em>-1 is prime, a <em>d</em>-valent graph of girth 6 having 2(<em>d</em><sup>2</sup>−<em>d</em>+1) vertices is exhibited. The method also gives the trivalent graph of girth 8 and 30 vertices.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 420-422"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80095-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87766700","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 elementary solution to a problem in restricted partitions","authors":"N. Metropolis, P.R. Stein","doi":"10.1016/S0021-9800(70)80091-6","DOIUrl":"10.1016/S0021-9800(70)80091-6","url":null,"abstract":"<div><p>The present paper deals with an apparently hitherto untreated problem in the theory of restricted partitions: What is the number <em>T<sub>n</sub><sup>(r)</sup></em> of distinct partitions of the composite integer <em>nr</em> that can be made by partwise addition of <em>n</em>—not necessarily distinct—partitions of <em>r</em>? The answer is given in the form of a finite series of binomial coefficients multiplied by certain integer coefficients which dependend only on <em>r</em>:</p><p><span><span><span><math><mrow><msubsup><mi>T</mi><mi>n</mi><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup><mo>=</mo><mstyle><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>r</mi><mo>−</mo><mi>g</mi><mo>−</mo><mn>1</mn></mrow></munderover><mrow><msubsup><mi>c</mi><mi>i</mi><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>g</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>g</mi><mo>+</mo><mi>i</mi></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>,</mo><mo>where</mo><mi>g</mi><mo>=</mo><mrow><mo>[</mo><mrow><mfrac><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow><mo>]</mo></mrow></mrow></mstyle></mrow></math></span></span></span></p><p>In general the non-vanishing <em>c<sub>i</sub><sup>(r)</sup></em> must be determined by direct calculation; in this paper we give them for all <em>r</em>≤11. Several other interpretations of <em>T<sub>n</sub><sup>(r)</sup></em> are given, and some additional open questions concerning the interpretation of the results are discussed.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 365-376"},"PeriodicalIF":0.0,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80091-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81638956","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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