Journal of Combinatorial Theory最新文献

{"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_m</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}

{"title":"Algebraic approach to stopping variable problems: Representation theory and applications","authors":"Melvin Tainiter","doi":"10.1016/S0021-9800(70)80022-9","DOIUrl":"10.1016/S0021-9800(70)80022-9","url":null,"abstract":"<div><p>We develop the relationship between distributive lattices and stopping variable problems by showing that the class of stopping variables has this structure. Using representation theory for distributive lattices we reduce the “secretary problem” and the <em>S_n/n</em>, problem for Bernoulli trials to linear programming problems.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 148-161"},"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)80022-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82155568","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":"Power-translation invariance in discrete groups—A characterization of finite Hamiltonian groups","authors":"William R. Emerson","doi":"10.1016/S0021-9800(70)80051-5","DOIUrl":"10.1016/S0021-9800(70)80051-5","url":null,"abstract":"<div><p>The primary result of this paper is the resolution of the question: Which non-Abelian discrete groups <em>G</em> satisfy (for some <em>n</em>&gt;1) |<em>(aS)ⁿ</em>|=|<em>Sⁿ</em>| for all <em>S G</em> and <em>a</em>∈<em>G</em>, (A_n) where |*| denotes the counting measure and <em>Sⁿ</em>={<em>s</em>₁…<em>s_n</em>:<em>s_i</em>∈<em>S</em>, 1≤<em>i</em>≤<em>n</em>}?</p><p>We prove that a discrete group <em>G</em> satisfies (A_n) for some integer <em>n</em>&gt;1 iff <em>G</em> is a finite Hamiltonian group. Furthermore, if γ denotes the invariant defined for finite Abelian groups introduced in [1] and <em>H</em> is any finite Hamiltonian group, then <em>H</em> satisfies (A_n) iff γ(<em>H′</em>)≥<em>n</em>, where <em>H′</em> denotes the unique (up to isomorphism) maximal Abelian subgroup of <em>H</em>. In the course of this development a number of results concerning finite Hamiltonian groups are obtained. We conclude with a section on related conditions as well as a discussion of the general locally compact case.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 6-26"},"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)80051-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73184034","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 note on coefficients of chromatic polynomials","authors":"V. Chvátal","doi":"10.1016/S0021-9800(70)80058-8","DOIUrl":"10.1016/S0021-9800(70)80058-8","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 95-96"},"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)80058-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84635832","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":"Decomposing partial orderings into chains","authors":"Kenneth P. Bogart","doi":"10.1016/S0021-9800(70)80059-X","DOIUrl":"10.1016/S0021-9800(70)80059-X","url":null,"abstract":"<div><p>Dilworth's theorem gives the number of chains whose union is a given partially ordered set in terms of intrinsic properties of the partial ordering. This paper uses Dilworth's theorem to find the number of chains needed to uniquely determine a given partially ordered set in terms of intrinsic properties of the partial ordering. If <em>a</em> covers <em>b</em> and <em>c</em> covers <em>d</em>, then (<em>a, b</em>) and (<em>c, d</em>) are <em>incomparable covers</em> if either <em>a</em> or <em>b</em> is incomparable with either <em>c</em> or <em>d</em>. We prove that the number of chains whose transtitive closure is a given partial ordering is the largest number of elements in any set of incomparable covers plus the number of isolated elements of the partially ordered set.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 97-99"},"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)80059-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86887213","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":"Hamiltonian circuits on 3-polytopes","authors":"David Barnette ,&nbsp;Ernest Jucovič","doi":"10.1016/S0021-9800(70)80054-0","DOIUrl":"10.1016/S0021-9800(70)80054-0","url":null,"abstract":"<div><p>The smallest number of vertices, edges, or faces of any 3-polytope with no Hamiltonian circuit is determined. Similar results are found for simplicial polytopes with no Hamiltonian circuit.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 54-59"},"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)80054-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78400187","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":"Zum basisproblem der nicht in die projektive ebene einbettbaren graphen, I","authors":"K. Wagner","doi":"10.1016/S0021-9800(70)80052-7","DOIUrl":"https://doi.org/10.1016/S0021-9800(70)80052-7","url":null,"abstract":"<div><p>Le <em>Γ</em> be the set of all finite graphs which are not embeddable into the projective plane. We write <em>G</em>₁≺<em>G</em>₂, if <em>G</em>₂ is <em>homomorphic</em> to <em>G</em>₁, that is, if there is a subgraph <em>G</em>′₂ of <em>G</em>₂ such that <em>G</em>₁ can be obtained from <em>G</em>′₂ by contraction of edges in <em>G</em>′₂. The subset of all minimal graphs in the partially ordered set <em>Γ</em>_≺ is called the <em>minimal basis</em> of <em>Γ</em>. We denote this basis by M(<em>Γ</em>). Every graph of M(<em>Γ</em>) is called a minimal graph (of <em>Γ</em>). In a former paper [2], we determined all disconnected minimal graphs and all minimal graphs separable by one or two vertices. Hence, we need only consider three-connected minimal graphs. Let us say that in our Proposition 7 of [2] we omitted a bracket in <em>G</em>₁₄. In this graph, namely, the uppermost 2 must be replaced by &lt;2&gt;.</p><p>In the following we deal with two new cases, in which we succeed in determining the minimal graphs:</p><p>First, let us suppose that <em>G</em>∈M(<em>Γ</em>) has the properties: (1) <em>G</em> is (at least) three-connected and (2) there is a vertex <em>a</em> in <em>G</em> such that the graph <em>G/a</em> (that is, one has to destroy the vertex <em>a</em> and all edges of <em>G</em> incident to <em>a</em>) contains a topologic <em>K</em>₅ but there exists no topologic <em>K</em>_3,3 contained in <em>G/a</em>. We prove the <em>theorem</em> that there are exactly two graphs in M(<em>Γ</em>) satisfying (1) and (2). One of these graphs is the <em>G</em>₁₄. The other graph, denoted by <em>D</em>, can be obtained from the <em>K</em>₅ where we let <em>T</em> be a triple of neighboring edges in the <em>K</em>₅, by inserting a new vertex on every edge of <em>T</em> and by then connecting these three new vertices through three edges with one further new vertex <em>a</em> lying outside the topologic <em>K</em>₅ (see Figure 2).</p><p>Second, we consider non-planar graphs <em>G</em> with the property that <em>G/a</em> is planar for every vertex <em>a</em> of <em>G</em>. In another paper [3] these graphs have been called <em>nearly planar</em>. Our next question is: <em>Are there nearly planar graphs not being embeddable into the projective plane?</em> We prove the <em>theorem</em> that all nearly planar graphs can be embedded into the projective plane with a single exception of one graph <em>F</em>. This exceptional graph <em>F</em> consists of two disjoint <em>K</em>₄ and four edges (<em>a_i, b_i</em>), <em>i</em>=1,…, 4 which join the vertices <em>a_i</em> of the one <em>K</em>₄ with the vertices <em>b_i</em> of the other <em>K</em>₄. We then show that this graph is a minimal graph of <e","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 27-43"},"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)80052-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137254637","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 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}

{"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}