D. Kleitman, A. Martin-Löf, B. Rothschild , A. Whinston
{"title":"A matching theorem for graphs","authors":"D. Kleitman, A. Martin-Löf, B. Rothschild , A. Whinston","doi":"10.1016/S0021-9800(70)80013-8","DOIUrl":"10.1016/S0021-9800(70)80013-8","url":null,"abstract":"<div><p>Let the vertices of an undirected graph be given labels 1, 2, …, <em>n</em>, 1′, 2′, …, <em>n</em>′ such that each vertex has at least <em>n</em>−1 different labels without both <em>i</em> and <em>i</em>′ for any <em>i</em>. Then among all paths between a vertex labeled <em>i</em> and a vertex labeled <em>i</em>′ for any <em>i</em>, the maximum number which are mutually edge disjoint equals the minimum size of an edge cut-set separating all vertices labeled <em>i</em> from all those labeled <em>i</em>′ for any <em>i</em>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 1","pages":"Pages 104-114"},"PeriodicalIF":0.0,"publicationDate":"1970-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80013-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77152728","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 distance between points in random trees","authors":"A. Meir, J.W. Moon","doi":"10.1016/S0021-9800(70)80012-6","DOIUrl":"10.1016/S0021-9800(70)80012-6","url":null,"abstract":"<div><p>Let <em>γ</em> denote the number of points in the path joining two arbitrary points in a random tree <em>T<sub>n</sub></em> with <em>n</em> labeled points. It is shown, among other things, that <em>E(γ)∼(1/2πn)</em><sup>1/2</sup>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 1","pages":"Pages 99-103"},"PeriodicalIF":0.0,"publicationDate":"1970-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80012-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86682278","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":"Toward a theory of crossing numbers","authors":"W.T. Tutte","doi":"10.1016/S0021-9800(70)80007-2","DOIUrl":"10.1016/S0021-9800(70)80007-2","url":null,"abstract":"<div><p>An algebraic structure, related to the crossing number, is constructed from a given graph.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 1","pages":"Pages 45-53"},"PeriodicalIF":0.0,"publicationDate":"1970-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80007-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84344304","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":"Heterogeneous algebras","authors":"Garrett Birkhoff, John D. Lipson","doi":"10.1016/S0021-9800(70)80014-X","DOIUrl":"https://doi.org/10.1016/S0021-9800(70)80014-X","url":null,"abstract":"<div><p>Many of the basic theorems about general “algebras” derived in [1, Ch. 6] are extended to a class of <em>heterogeneous algebras</em> which includes automata, state machines, and monoids acting on sets. It is shown that some algebras can be fruitfully studied, using different interpretations, both as (homogeneous) algebras and as heterogeneous algebras, and a non-trivial “free machine” is constructed as an application. The extent of the overlap with previous work of Higgins [9] is specified.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 1","pages":"Pages 115-133"},"PeriodicalIF":0.0,"publicationDate":"1970-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80014-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137290738","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":"Connectivity of transitive graphs","authors":"Mark E. Watkins","doi":"10.1016/S0021-9800(70)80005-9","DOIUrl":"10.1016/S0021-9800(70)80005-9","url":null,"abstract":"<div><p>The conditions imposed by edge-transitivity and vertex-transitivity on the connectivity of simple graphs are investigated. Particular attention is given to the structure of those vertex-transitive graphs for which the degree of regularity exceeds the connectivity.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 1","pages":"Pages 23-29"},"PeriodicalIF":0.0,"publicationDate":"1970-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80005-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74773921","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":"Designs derived from permutation groups","authors":"Marshall Hall Jr., Richard Lane, David Wales","doi":"10.1016/S0021-9800(70)80004-7","DOIUrl":"10.1016/S0021-9800(70)80004-7","url":null,"abstract":"<div><p>Let <em>G</em> be a transitive permutation group on a set Ω of <em>v</em> points {1, 2, …, <em>v</em>}. Let <em>H</em> be an intransitive subgroup of <em>G</em> and let Δ a set of <em>k</em> points where <em>Δ</em> consists of complete orbits of <em>H</em>. Then the images <em>Δ</em><sup>x</sup> of <em>Δ</em> under permutations <em>x</em> of <em>Δ</em> have been shown by the first author to be a partially balanced block design <em>D</em> with <em>G</em> as a group of automorphisms. Under certain circumstances <em>D</em> is a balanced incomplete block design. Here a representation of the simple group PSL<sub>3</sub>(4) of order 20,160 on 56 letters leads to a new symmetric block design with parameters <em>v</em>=56, <em>k</em>-11, <em>λ</em>=2. A representation of the simple group of order 25,920 as U<sub>4</sub>(4) on 45 isotropic points gives a symmetric design with <em>v</em>=45, <em>k</em>=12, <em>λ</em>=3. One representation of U<sub>4</sub>(4) on 40 points, gives the design of planes in PG(3, 3) and exhibits the isomorphism of this group to the symplectic group S<sub>4</sub>(3).</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 1","pages":"Pages 12-22"},"PeriodicalIF":0.0,"publicationDate":"1970-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80004-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75491352","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":"Remarks on the van der Waerden permanent conjecture","authors":"Andrew M. Gleason","doi":"10.1016/S0021-9800(70)80008-4","DOIUrl":"10.1016/S0021-9800(70)80008-4","url":null,"abstract":"<div><p>The van der Waerden permanent conjecture is shown to belong to a large family of conjectured inequalities which are of some interest in themselves and all of which <em>might</em> be provable by a routine computation with convex bodies. In fact, the permanent conjecture in cases <em>n</em>=3 and 4 does yield to this method. For <em>n</em>=5, with the computations made by hand, no proof was found, but a slight extension of the computation (which would probably require electronic assistance) could still settle this case and perhaps even a few more small values of <em>n</em>. The question of whether the method must work remains open.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 1","pages":"Pages 54-64"},"PeriodicalIF":0.0,"publicationDate":"1970-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80008-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78678916","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":"Probability of a pure equilibrium point in n-person games","authors":"Melvin Dresher","doi":"10.1016/S0021-9800(70)80015-1","DOIUrl":"10.1016/S0021-9800(70)80015-1","url":null,"abstract":"<div><p>A “random” <em>n</em>-person non-cooperative game—the game that prohibits communication and therefore coalitions among the <em>n</em> players—is shown to have with high probability a pure strategy solution. Such a solution is by definition an equilibrium point or a set of strategies, one for each player, such that if <em>n</em>−1 players use their equilibrium strategies then the <em>n</em>-th player has no reason to deviate from his equilibrium strategy. It is shown that the probability of a solution in pure strategies for large random <em>n</em>-person games converges to (1−1/<em>e</em>) for all <em>n</em>≥2.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 1","pages":"Pages 134-145"},"PeriodicalIF":0.0,"publicationDate":"1970-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80015-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89073303","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 lower bound for the number of non-isomorphic matroids","authors":"Béla Bollobás","doi":"10.1016/S0021-9800(69)80064-5","DOIUrl":"https://doi.org/10.1016/S0021-9800(69)80064-5","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 4","pages":"Pages 366-368"},"PeriodicalIF":0.0,"publicationDate":"1969-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(69)80064-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91610975","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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