{"title":"On graphs with the same restricted U-polynomial and the U-polynomial for rooted graphs","authors":"José Aliste-Prieto , José Zamora , Anna de Mier","doi":"10.1016/j.endm.2018.06.032","DOIUrl":"10.1016/j.endm.2018.06.032","url":null,"abstract":"<div><p>In this abstract, we construct explicitly, for every <em>k</em>, pairs of non-isomorphic trees with the same restricted <em>U</em>-polynomial; by this we mean that the polynomials agree on terms with degree at most <em>k</em>. The construction is done purely in algebraic terms, after introducing and studying a generalization of the <em>U</em>-polynomial to rooted graphs.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.032","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128554122","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":"Dyck-Eulerian digraphs","authors":"Pietro Codara , Ottavio M. D'Antona","doi":"10.1016/j.endm.2018.06.041","DOIUrl":"10.1016/j.endm.2018.06.041","url":null,"abstract":"<div><p>We introduce a family of Eulerian digraphs, <span><math><mi>E</mi></math></span>, associated with Dyck words. We provide the algorithms implementing the bijection between <span><math><mi>E</mi></math></span> and <span><math><mi>W</mi></math></span>, the set of Dyck words. To do so, we exploit a binary matrix, that we call <em>Dyck matrix</em>, representing the cycles of an Eulerian digraph.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.041","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124985359","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":"A characterization of weight-regular partitions of graphs","authors":"Aida Abiad","doi":"10.1016/j.endm.2018.06.050","DOIUrl":"10.1016/j.endm.2018.06.050","url":null,"abstract":"<div><p>A partition <span><math><mi>P</mi><mo>=</mo><mo>{</mo><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>m</mi></mrow></msub><mo>}</mo></math></span> of the vertex set <em>V</em> of a graph is regular if, for all <em>i</em>, <em>j</em>, the number of neighbors which a vertex in <em>V</em><sub><em>i</em></sub> has in the set <em>V</em><sub><em>j</em></sub> is independent of the choice of vertex in <em>V</em><sub><em>i</em></sub>. The natural generalization of a regular partition, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex <span><math><mi>u</mi><mo>∈</mo><mi>V</mi></math></span> a weight which equals the corresponding entry <span><math><msub><mrow><mi>ν</mi></mrow><mrow><mi>u</mi></mrow></msub></math></span> of the Perron eigenvector <strong><em>ν</em></strong>. In this work we investigate when a weight-regular partition of a graph is regular in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we provide a new characterization of weight-regular partitions by using a Hoffman-like polynomial.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.050","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122608715","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":"On the discrete Brunn-Minkowski inequality by Gradner&Gronchi","authors":"David Iglesias","doi":"10.1016/j.endm.2018.06.051","DOIUrl":"10.1016/j.endm.2018.06.051","url":null,"abstract":"<div><p>In 2002 Gardner and Gronchi obtained a discrete analogue of the Brunn-Minkowski inequality. They proved that for finite subsets <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mi>dim</mi><mo></mo><mi>B</mi><mo>=</mo><mi>n</mi></math></span>, the inequality <span><math><mo>|</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo>|</mo><mo>≥</mo><mo>|</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mo>|</mo><mi>A</mi><mo>|</mo></mrow><mrow><mi>B</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mo>|</mo><mi>B</mi><mo>|</mo></mrow><mrow><mi>B</mi></mrow></msubsup><mo>|</mo></math></span> holds, where <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mo>|</mo><mi>A</mi><mo>|</mo></mrow><mrow><mi>B</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mo>|</mo><mi>B</mi><mo>|</mo></mrow><mrow><mi>B</mi></mrow></msubsup></math></span> are particular subsets of the integer lattice, called <em>B</em>-initial segments. The aim of this paper is to provide a method in order to compute <span><math><mo>|</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mo>|</mo><mi>A</mi><mo>|</mo></mrow><mrow><mi>B</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mo>|</mo><mi>B</mi><mo>|</mo></mrow><mrow><mi>B</mi></mrow></msubsup><mo>|</mo></math></span> and so, to implement this inequality.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.051","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132003594","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":"Graph Operations Preserving W2-Property","authors":"Vadim E. Levit, Eugen Mandrescu","doi":"10.1016/j.endm.2018.06.007","DOIUrl":"10.1016/j.endm.2018.06.007","url":null,"abstract":"<div><p>A graph is <em>well-covered</em> if all its maximal independent sets are of the same size (Plummer, 1970). A graph <em>G</em> belongs to class <strong>W</strong><sub><em>n</em></sub> if every <em>n</em> pairwise disjoint independent sets in <em>G</em> are included in <em>n</em> pairwise disjoint maximum independent sets (Staples, 1975). Clearly, <strong>W</strong><sub>1</sub> is the family of all well-covered graphs. Staples showed a number of ways to build graphs in <strong>W</strong><sub><em>n</em></sub>, using graphs from <strong>W</strong><sub><em>n</em></sub> or <strong>W</strong><sub><em>n</em>+1</sub>. In this paper, we construct some more infinite subfamilies of the class <strong>W</strong><sub><strong>2</strong></sub> by means of corona, join, and rooted product of graphs.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133383754","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, 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":null,"pages":null},"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, Yoshimi Egawa, 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":null,"pages":null},"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}
Elena Cabrera Revuelta, María José Chávez de Diego , Alberto Márquez Pérez
{"title":"The Fortress Problem in Terms of the Number of Reflex and Convex Vertices. A 3D objects scanning application","authors":"Elena Cabrera Revuelta, María José Chávez de Diego , Alberto Márquez Pérez","doi":"10.1016/j.endm.2018.06.030","DOIUrl":"10.1016/j.endm.2018.06.030","url":null,"abstract":"<div><p>This work focuses on the visibility of exterior of a polygon. We show a lower bound for the Fortress Problem applied to polygons <em>P</em> having <em>n</em> vertices in terms of the number of reflex vertices, the number of convex vertices and the number of pockets that are found when the convex hull is made on <em>P</em>. The results are related to the task of the geometric data acquisition for architectural surveys through techniques such as laser scanner.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.030","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129103490","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}
Javier Cilleruelo , Oriol Serra , Maximilian Wötzel
{"title":"Set systems with distinct sumsets","authors":"Javier Cilleruelo , Oriol Serra , Maximilian Wötzel","doi":"10.1016/j.endm.2018.06.004","DOIUrl":"10.1016/j.endm.2018.06.004","url":null,"abstract":"<div><p>A family <span><math><mi>A</mi></math></span> of <em>k</em>-subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi><mo>}</mo></math></span> is a Sidon system if the sumsets <span><math><mi>A</mi><mo>+</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></math></span> are pairwise distinct. We show that the largest cardinality <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> of a Sidon system of <em>k</em>-subsets of <span><math><mo>[</mo><mi>N</mi><mo>]</mo></math></span> satisfies <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>N</mi><mo>−</mo><mi>k</mi></math></span> and the asymptotic lower bound <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span>. More precise bounds on <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> are obtained for <span><math><mi>k</mi><mo>≤</mo><mn>3</mn></math></span>. We also obtain the threshold probability for a random system to be Sidon for <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> and 3.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127014072","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":"On a problem of Sárközy and Sós for multivariate linear forms","authors":"Juanjo Rué , 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":null,"pages":null},"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}