{"title":"On the factorisation of the complete graph into factors of diameter 2","authors":"Norbert Sauer","doi":"10.1016/S0021-9800(70)80096-5","DOIUrl":"10.1016/S0021-9800(70)80096-5","url":null,"abstract":"<div><p><em>f(k)</em> denotes the smallest number <em>n</em> such that the complete graph (<em>n</em>) can be decomposed into <em>k</em> factors of diameter 2. So far the following results have been obtained [1]:</p><p><span><span><span><math><mrow><mn>4</mn><mi>k</mi><mo>−</mo><mn>1</mn><mo>⩽</mo><mi>f</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mo>⩽</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><mn>6</mn><mi>k</mi><mo>−</mo><mn>7</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>2</mn><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow></math></span></span></span></p><p><em>f</em>(2)≤5, <em>f</em>(3)≤13, <em>f</em>(4)≤41, <em>f</em>(5)≤71, <em>f</em>(6)≤157, <em>f</em>(7)≤193, <em>f</em>(8)≤193, <em>f</em>(9)≤379, <em>f</em>(10)≤521 and there exists a positive integer <em>K</em> such that for any integer <em>k>K</em>:</p><p><span><span><span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mo>⩽</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>49</mn></mrow><mrow><mn>10</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><msup><mi>k</mi><mn>2</mn></msup><mo>log</mo><mo></mo><mi>k</mi></mrow></math></span></span></span></p><p>The purpose of this paper is to improve the upper bound on <em>f(k)</em> by showing that <em>f(k)≤7k</em> holds.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 423-426"},"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)80096-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73969321","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 n-uniform Hjelmslev planes","authors":"David A. Drake","doi":"10.1016/S0021-9800(70)80066-7","DOIUrl":"10.1016/S0021-9800(70)80066-7","url":null,"abstract":"<div><p>We define 1-<em>uniform</em> and <em>strongly</em> 1-<em>uniform</em> Hjelmslev planes (<em>H</em>-planes) to be the ordinary affine and projective planes. An <em>n-uniform H</em>-plane (<em>n</em>>1) is an <em>H</em>-plane whose point neighborhoods all are (<em>n</em>−1)-uniform affine <em>H</em>-planes. A <em>strongly n-uniform H</em>-plane (<em>n</em>>1) is an <em>n</em>-uniform projective <em>H</em>-plane which collapses to a strongly (<em>n</em>−1)-uniform <em>H</em>-plane upon identifying maximally connected points (points joined by <em>t</em> lines). All uniform projective <em>H</em>-planes are strongly <em>n</em>-uniform with <em>n</em>=1 or <em>n</em>=2. It is proved that all Desarguesian projective <em>H</em>-planes are strongly <em>n</em>-uniform. Many nice intersection properties are given for <em>n</em>-uniform <em>H</em>-planes; strongly <em>n</em>-uniform <em>H</em>-planes satisfy a strong intersection property called “property <em>A</em>.” It is proved that an <em>n</em>-uniform projective <em>H</em>-plane <em>π</em> is strongly <em>n</em>-uniform if and only if <em>π</em> satisfies property <em>A</em>, and also if and only if <em>π</em><sup>*</sup>, the dual of <em>π</em>, is <em>n</em>-uniform.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 267-288"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80066-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87489287","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":"Disproof of a conjecture of Erdös and moser on tournaments","authors":"K.B. Reid , E.T. Parker","doi":"10.1016/S0021-9800(70)80061-8","DOIUrl":"10.1016/S0021-9800(70)80061-8","url":null,"abstract":"<div><p>Erdös and Moser [1] displayed a tournament of order 7 with no transitive subtournament of order 4 and conjectured for each positive integer <em>k</em> existence of a tournament of order 2<sup><em>k</em>−1</sup>−1 with no transitive subtournament of order <em>k</em>. The conjecture is disproved for <em>k</em>=5. Further, every tournament of order 14 has a transitive subtournament of order 5. Inductively, the conjecture is false for all orders above 5. Existence and uniqueness of a tournament of order 13 having no transitive subtournament of order 5 are shown.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 225-238"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80061-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76399349","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":"n-Hamiltonian graphs","authors":"Gary Chartrand , S.F. Kapoor, Don R. Lick","doi":"10.1016/S0021-9800(70)80069-2","DOIUrl":"10.1016/S0021-9800(70)80069-2","url":null,"abstract":"<div><p>A graph <em>G</em> with <em>p</em>≥3 points, 0≤<em>n</em>≤<em>p</em>−3, is called <em>n</em>-Hamiltonian if the removal of any <em>k</em> points from <em>G</em>, 0≤<em>k≤n</em>, results in a Hamiltonian graph. This generalizes the concept of Hamiltonian graphs in as much as the 0-Hamiltonian graphs are precisely the Hamiltonian graphs. Sufficient conditions for a graph to be <em>n</em>-Hamiltonian are presented, including generalizations of results on Hamiltonian graphs due to Dirac, Ore, and Pósa.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 308-312"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80069-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90438509","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":"Algebras defined by patterns of zeros","authors":"Robert L. Davis","doi":"10.1016/S0021-9800(70)80064-3","DOIUrl":"10.1016/S0021-9800(70)80064-3","url":null,"abstract":"<div><p>If <em>X</em> is any set and <em>T</em> any subset of <em>X×X</em>, call <em>V(T)</em> the vector space of all functions with support <em>T</em> and values in a field of characteristic zero. The main theorem below shows that a necessary and sufficient condition that <em>V(T)</em> admit (and be closed under) a convolution <em>f*g(x,y)=Σf(x, z) g(z, y)</em>, sum over all <em>z∈X</em>, is that <em>T</em> be a locally finite transitive relation. One special corollary is that, if <em>V(T)</em> consists of upper triangular (finite or infinite) matrices and contains the identity, then there is such a convolution if and only if <em>V(T)</em> is the incidence algebra, as defined by Rota, of the locally finite partial order <em>T</em>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 257-260"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80064-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88266508","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":"Transformations on Directed graphs","authors":"Sanat K. Basu","doi":"10.1016/S0021-9800(70)80063-1","DOIUrl":"10.1016/S0021-9800(70)80063-1","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 244-256"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80063-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73308089","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 with circulant adjacency matrices","authors":"Bernard Elspas, James Turner","doi":"10.1016/S0021-9800(70)80068-0","DOIUrl":"10.1016/S0021-9800(70)80068-0","url":null,"abstract":"<div><p>Properties of a graph (directed or undirected) whose adjacency matrix is a circulant are studied. Examples are given showing that the connection set determined by the first row of such a matrix need not be multiplicatively related to the connection set of an isomorphic graph. Two different criteria are given under which two graphs with circulant adjacency matrices are isomorphic if and only if their connection sets are multiplicatively related. The first criterion is that the graphs have a prime number of vertices. The second criterion is that the adjacency matrices have non-repeated eigenvalues. The final section gives a partial characterization of graphs with <em>n</em> vertices whose automorphism group is the cyclic group <em>C<sub>n</sub></em>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 297-307"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80068-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74252419","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 chromatic polynomials and the golden ratio","authors":"W.T. Tutte","doi":"10.1016/S0021-9800(70)80067-9","DOIUrl":"10.1016/S0021-9800(70)80067-9","url":null,"abstract":"<div><p>Let <em>M</em> be a triangulation of the 2-sphere, with <em>k</em> vertices. Let <em>P</em>(<em>M, n</em>) be its chromatic polynomial with respect to vertex-colorings. Then<span><span><span><math><mrow><mo>|</mo><mi>P</mi><mo>(</mo><mi>M</mi><mo>,</mo><mn>1</mn><mo>+</mo><mi>τ</mi><mo>)</mo><mo>|</mo><mo>⩽</mo><msup><mi>τ</mi><mrow><mn>5</mn><mo>−</mo><mi>k</mi></mrow></msup></mrow></math></span></span></span>where <em>τ</em> is the “golden ratio” <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msqrt><mn>5</mn></msqrt><mo>)</mo><mo>/</mo><mn>2</mn></mrow></math></span></p><p>This result is offered as a theoretical explanation of the empirical observation that <em>P</em>(<em>M, n</em>) tends to have a zero near <em>n</em>=1+<em>τ</em> (see [1]).</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 289-296"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80067-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75080945","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":"Group transitivity and a multiplicative function of a partition","authors":"H.O. Foulkes","doi":"10.1016/S0021-9800(70)80065-5","DOIUrl":"10.1016/S0021-9800(70)80065-5","url":null,"abstract":"<div><p>It is shown that a numerical function introduced by Tsuzuku in connection with group transitivity is related to exponential polynomials, and when considered as a function of a partition has multiplicative properties.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 261-266"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80065-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75658099","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 2-level orthogonal arrays of odd index","authors":"J.R. Blum , J.A. Schatz , E. Seiden","doi":"10.1016/S0021-9800(70)80062-X","DOIUrl":"10.1016/S0021-9800(70)80062-X","url":null,"abstract":"<div><p>In this paper we show that the maximum possible number of constraints for a 2-level orthogonal array of odd index with strength <em>t</em> is <em>t</em>+1.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 239-243"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80062-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75804294","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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