Journal of Combinatorial Theory最新文献

{"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":"The lexicographically least de Bruijn cycle","authors":"Harold Fredricksen","doi":"10.1016/S0021-9800(70)80050-3","DOIUrl":"10.1016/S0021-9800(70)80050-3","url":null,"abstract":"<div><p>We investigate Ford's method for generating a de Bruijn cycle of degree <em>n</em>. We show that the feedback function which generates the cycle has the minimum possible number of positions equal to 1 and we give an algorithm which finds those positions for all <em>n</em>.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 1-5"},"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)80050-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78854428","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 smallest triangle-free 4-chromatic 4-regular graph","authors":"V. Chvátal","doi":"10.1016/S0021-9800(70)80057-6","DOIUrl":"10.1016/S0021-9800(70)80057-6","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 93-94"},"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)80057-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73135958","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 completely unambiguous 5-polyhedral graph","authors":"David Barnette","doi":"10.1016/S0021-9800(70)80053-9","DOIUrl":"10.1016/S0021-9800(70)80053-9","url":null,"abstract":"<div><p>A 5-dimensional convex polytope <em>P</em> is constructed whose graph <em>G</em> has the property that if it is the graph a convex polytope <em>P</em>′ then <em>P</em>′ is combinatorially equivalent to <em>P</em> and, furthermore, <em>G</em> can be realized as the graph of <em>P</em> in only one way (i.e., if a subgraph of <em>G</em> determines a face of <em>P</em> it also determines a face of <em>P</em>′).</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 44-53"},"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)80053-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80273210","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 combinatorial invariant for finite Abelian groups with various applications","authors":"W.R. Emerson","doi":"10.1016/S0021-9800(70)80056-4","DOIUrl":"10.1016/S0021-9800(70)80056-4","url":null,"abstract":"<div><p>If <em>G</em> is any finite Abelian group define<span><span><span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle><munder><mo>∑</mo><mi>i</mi></munder><mrow><mo>(</mo><msub><mi>e</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mstyle></mrow></math></span></span></span></p><p>where <em>e_i</em> are the canonic invariants of <em>G</em> (<em>e</em>_i+1|<em>e</em>_i for all <em>i</em>). The primary result is the “super-additivity” of <em>γ</em>, i.e.,<span><span><span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>⩾</mo><mi>γ</mi><mrow><mo>(</mo><mrow><mrow><mi>G</mi><mo>/</mo><mi>H</mi></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mi>γ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>for</mo><mo>all</mo><mo>subgroups</mo><mo>H</mo><mo>of</mo><mo>G</mo><mo>.</mo><mrow><mo>(</mo><mo>∗</mo><mo>)</mo></mrow></mrow></math></span></span></span></p><p>In the process of establishing (*) the structure of Abelian groups is studied in great detail and a technique is developed for proving inequalities analogous to (*) for other invariants than γ.</p><p>We then apply (*) to obtain various further results of a combinatorial nature.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 72-92"},"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)80056-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80058666","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}