{"title":"Constructing generalized Heffter arrays via near alternating sign matrices","authors":"L. Mella , T. Traetta","doi":"10.1016/j.jcta.2024.105873","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>S</em> be a subset of a group <em>G</em> (not necessarily abelian) such that <span><math><mi>S</mi><mspace></mspace><mo>∩</mo><mo>−</mo><mi>S</mi></math></span> is empty or contains only elements of order 2, and let <span><math><mi>h</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and <span><math><mi>k</mi><mo>=</mo><mo>(</mo><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>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A <em>generalized Heffter array</em> GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over <em>G</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix <span><math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></math></span> such that: the <em>i</em>-th row (resp. <em>j</em>-th column) of <em>A</em> contains exactly <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (resp. <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>) nonzero elements, and the list <span><math><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>}</mo></math></span> equals <em>λ</em> times the set <span><math><mi>S</mi><mspace></mspace><mo>∪</mo><mspace></mspace><mo>−</mo><mi>S</mi></math></span>. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of <em>A</em> sums to zero (resp. a nonzero element), with respect to some ordering.</p><p>In this paper, we use <em>near alternating sign matrices</em> to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to 0 modulo 4. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular (<span><math><mi>m</mi><mo>≠</mo><mi>n</mi></math></span>) and with less than <em>n</em> nonzero entries in each row. Furthermore, we build nonzero sum GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over an arbitrary group <em>G</em> whenever <em>S</em> contains enough noninvolutions, thus extending previous nonconstructive results where <span><math><mo>±</mo><mi>S</mi><mo>=</mo><mi>G</mi><mo>∖</mo><mi>H</mi></math></span> for some subgroup <em>H</em> of <em>G</em>.</p><p>Finally, we describe how GHAs can be used to build orthogonal decompositions and biembeddings of Cayley graphs (over groups not necessarily abelian) onto orientable surfaces.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524000128","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let S be a subset of a group G (not necessarily abelian) such that is empty or contains only elements of order 2, and let and . A generalized Heffter array GHA over G is an matrix such that: the i-th row (resp. j-th column) of A contains exactly (resp. ) nonzero elements, and the list equals λ times the set . We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of A sums to zero (resp. a nonzero element), with respect to some ordering.
In this paper, we use near alternating sign matrices to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to 0 modulo 4. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular () and with less than n nonzero entries in each row. Furthermore, we build nonzero sum GHA over an arbitrary group G whenever S contains enough noninvolutions, thus extending previous nonconstructive results where for some subgroup H of G.
Finally, we describe how GHAs can be used to build orthogonal decompositions and biembeddings of Cayley graphs (over groups not necessarily abelian) onto orientable surfaces.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.