Electronic Notes in Discrete Mathematics最新文献

{"title":"On a problem of Sárközy and Sós for multivariate linear forms","authors":"Juanjo Rué ,&nbsp;Christoph Spiegel","doi":"10.1016/j.endm.2018.06.018","DOIUrl":"10.1016/j.endm.2018.06.018","url":null,"abstract":"<div><p>We prove that for pairwise co-prime numbers <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>≥</mo><mn>2</mn></math></span> there does not exist any infinite set of positive integers <span><math><mi>A</mi></math></span> such that the representation function <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>#</mi><mo>{</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>:</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>}</mo></math></span> becomes constant for <em>n</em> large enough. This result is a particular case of our main theorem, which poses a further step towards answering a question of Sárközy and Sós and widely extends a previous result of Cilleruelo and Rué for bivariate linear forms (Bull. of the London Math. Society 2009).</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 101-106"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125854341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"2-closed abelian permutation groups","authors":"Mariusz Grech,&nbsp;Andrzej Kisielewicz","doi":"10.1016/j.endm.2018.06.015","DOIUrl":"10.1016/j.endm.2018.06.015","url":null,"abstract":"<div><p>In this paper we demonstrate that the result by Zelikovskij concerning Königs problem for abelian permutation groups, reported in a recent survey, is false. We propose in this place two results on 2-closed abelian permutation groups which concern the same topic in a more general setting.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 83-88"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127854687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Edges incident with a vertex of degree greater than four and a lower bound on the number of contractible edges in a 4-connected graph","authors":"Shunsuke Nakamura,&nbsp;Yoshimi Egawa,&nbsp;Keiko Kotani","doi":"10.1016/j.endm.2018.06.005","DOIUrl":"10.1016/j.endm.2018.06.005","url":null,"abstract":"<div><p>In this paper, we prove that the number of 4-contractible edges (edges that after contraction do not change the connectivity of the initial graph) of a 4-connected graph <em>G</em> is at least <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mn>28</mn><mo>)</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>x</mi><mo>∈</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>≥</mo><mn>5</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><msub><mrow><mi>deg</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>⁡</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>V</mi></mrow><mrow><mo>≥</mo><mn>5</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes the set of those vertices of <em>G</em> which have degree greater than or equal to 5.</p><p>This is the refinement of the result proved by Ando et al. [On the number of 4-contractible edges in 4-connected graphs, <em>J. Combin. Theory Ser. B</em> <strong>99</strong> (2009) 97–109].</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 23-28"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128571251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The complete classification of empty lattice 4-simplices","authors":"Óscar Iglesias Valiño ,&nbsp;Francisco Santos","doi":"10.1016/j.endm.2018.06.027","DOIUrl":"https://doi.org/10.1016/j.endm.2018.06.027","url":null,"abstract":"<div><p>In previous work we classified all empty 4-simplices of width at least three. We here classify those of width two. There are 2 two-parameter families that project to the second dilation of a unimodular triangle, <span><math><mn>29</mn><mo>+</mo><mn>23</mn></math></span> one-parameter families of them that project to hollow 3-polytopes, and 2282 individual ones that do not.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 155-160"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138362561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Iterated Sumsets and Olson's Generalization of the Erdős-Ginzburg-Ziv Theorem","authors":"David J. Grynkiewicz","doi":"10.1016/j.endm.2018.06.006","DOIUrl":"10.1016/j.endm.2018.06.006","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a finite abelian group with &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. The Kemperman Structure Theorem characterizes all subsets &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; satisfying &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and has been extended to cover the case when &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Utilizing these results, we provide a precise structural description of all finite subsets &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; when &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; (also when &lt;em&gt;G&lt;/em&gt; is infinite), in which case many of the pathological possibilities from the case &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; vanish, particularly for large &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. The structural description is combined with other arguments to generalize a subsequence sum result of Olson asserting that a sequence &lt;em&gt;S&lt;/em&gt; of terms from &lt;em&gt;G&lt;/em&gt; having length &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; must either have every element of &lt;em&gt;G&lt;/em&gt; representable as a sum of &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-terms from &lt;em&gt;S&lt;/em&gt; or else have all but &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; of its terms lying in a common &lt;em&gt;H&lt;/em&gt;-coset for some &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We show that the much weaker hypothesis &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; suffices to obtain a nearly identical conclusion, where for the case &lt;em&gt;H&lt;/em&gt; is trivial we must allow all but &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; terms of &lt;em&gt;S&lt;/em&gt; to be from the same &lt;em&gt;H&lt;/em&gt;-coset. The bound on &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is improved for sever","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 29-34"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123742548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extended Lagrange's four-square theorem","authors":"J. Lacalle,&nbsp;L.N. Gatti","doi":"10.1016/j.endm.2018.06.036","DOIUrl":"10.1016/j.endm.2018.06.036","url":null,"abstract":"<div><p>We prove the following extension of Lagrange's theorem: given a prime number <em>p</em> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>,</mo><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>3</mn></math></span>, such that <span><math><msup><mrow><mo>∥</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∥</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>p</mi></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span> and <span><math><mo>〈</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>〉</mo><mo>=</mo><mn>0</mn></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo>≤</mo><mi>k</mi></math></span>, then there exists <span><math><mi>v</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> such that <span><math><mo>〈</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mi>v</mi><mo>〉</mo><mo>=</mo><mn>0</mn></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span> and<span><span><span><math><mo>∥</mo><mi>v</mi><mo>∥</mo><mo>=</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>=</mo><mi>p</mi></math></span></span></span> This means that, in <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>, any system of orthogonal vectors of norm <em>p</em> can be completed to a base. We conjecture that the result holds for every norm <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 209-214"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.036","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134166636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An improvement of the lower bound on the maximum number of halving lines in planar sets with 32 points","authors":"Javier Rodrigo,&nbsp;Ma Dolores López","doi":"10.1016/j.endm.2018.06.052","DOIUrl":"10.1016/j.endm.2018.06.052","url":null,"abstract":"<div><p>In this paper we give a recursive lower bound on the maximum number of halving lines for sets in the plane and as a consequence we improve the current best lower bound on the maximum number of halving lines for sets in the plane with 32 points.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 305-310"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.052","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116093958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sums of finite subsets in Rd","authors":"Mario Huicochea","doi":"10.1016/j.endm.2018.06.012","DOIUrl":"10.1016/j.endm.2018.06.012","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be nonempty finite subsets of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> not contained in an affine hyperplane for each <span><math><mi>i</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></math></span>. First we get a sharp lower bound on <span><math><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo></math></span> when <span><math><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>=</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>. Using this result and other ideas, we find a nontrivial lower bound on <span><math><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>|</mo></math></span> which generalizes a result of M. Matolcsi and I. Z. Ruzsa [7].</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 65-69"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113938794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Structure and Enumeration of K4-minor-free links and link diagrams","authors":"Juanjo Rué ,&nbsp;Dimitrios M. Thilikos ,&nbsp;Vasiliki Velona","doi":"10.1016/j.endm.2018.06.021","DOIUrl":"10.1016/j.endm.2018.06.021","url":null,"abstract":"<div><p>We study the class <span><math><mi>L</mi></math></span> of link types that admit a K₄-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K₄. We prove that <span><math><mi>L</mi></math></span> is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate <span><math><mi>L</mi></math></span> and subclasses of it, with respect to the minimal number of crossings or edges in a projection of <span><math><mi>L</mi><mo>∈</mo><mi>L</mi></math></span>. Further, we enumerate (both exactly and asymptotically) all connected K₄-minor-free link diagrams, all minimal connected K₄-minor-free link diagrams, and all K₄-minor-free diagrams of the unknot.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 119-124"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128307561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometric quadrangulations of a polygon","authors":"Atsuhiro Nakamoto ,&nbsp;Gen Kawatani,&nbsp;Naoki Matsumoto,&nbsp;Jorge Urrutia","doi":"10.1016/j.endm.2018.06.011","DOIUrl":"10.1016/j.endm.2018.06.011","url":null,"abstract":"<div><p>We deal with a <em>geometric quadrangulation</em> of a polygon <em>P</em>. We define a new notion called “the spirality” of <em>P</em>, which measures how close <em>P</em> is from being a convex polygon. Using the spirality, we describe (1) a condition of <em>P</em> to admit a geometric quadrangulation, and (2) a condition of <em>P</em> guaranteeing that any two geometric quadrangulations on <em>P</em> can be related by a sequence of edge flips.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"68 ","pages":"Pages 59-64"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131704617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}