Discrete Mathematics最新文献

{"title":"A digraph version of the Friendship Theorem","authors":"Myungho Choi,&nbsp;Hojin Chu,&nbsp;Suh-Ryung Kim","doi":"10.1016/j.disc.2024.114238","DOIUrl":"10.1016/j.disc.2024.114238","url":null,"abstract":"<div><p>The Friendship Theorem states that if in a party any pair of persons has precisely one common friend, then there is always a person who is everybody's friend and the theorem has been proved by Paul Erdős, Alfréd Rényi, and Vera T. Sós in 1966. “What would happen if instead any pair of persons likes precisely one person?” While a friendship relation is symmetric, a liking relation may not be symmetric. Therefore to represent a liking relation we should use a directed graph. We call this digraph a “liking digraph”. It is easy to check that a symmetric liking digraph becomes a friendship graph if each directed cycle of length two is replaced with an edge. In this paper, we provide a digraph formulation of the Friendship Theorem which characterizes the liking digraphs. We also establish a sufficient and necessary condition for the existence of liking digraphs.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114238"},"PeriodicalIF":0.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003698/pdfft?md5=368b7d4c1379f8549152a904b901804b&pid=1-s2.0-S0012365X24003698-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extremal graphs for the odd prism","authors":"Xiaocong He ,&nbsp;Yongtao Li ,&nbsp;Lihua Feng","doi":"10.1016/j.disc.2024.114249","DOIUrl":"10.1016/j.disc.2024.114249","url":null,"abstract":"<div><p>The Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> of a graph <em>H</em> is the maximum number of edges in an <em>n</em>-vertex graph which does not contain <em>H</em> as a subgraph. The Turán number of regular polyhedrons was widely studied in a series of works due to Simonovits. In this paper, we shall present the exact Turán number of the prism <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>□</mo></mrow></msubsup></math></span>, which is defined as the Cartesian product of an odd cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> and an edge <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Applying a deep theorem of Simonovits and a stability result of Yuan (2022) <span><span>[55]</span></span>, we shall determine the exact value of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>□</mo></mrow></msubsup><mo>)</mo></math></span> for every <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and sufficiently large <em>n</em>, and we also characterize the extremal graphs. Moreover, in the case of <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>, motivated by a recent result of Xiao et al. (2022) <span><span>[49]</span></span>, we will determine the exact value of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow><mrow><mo>□</mo></mrow></msubsup><mo>)</mo></math></span> for every <em>n</em> instead of for sufficiently large <em>n</em>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114249"},"PeriodicalIF":0.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003807/pdfft?md5=e2bc8fb4249126377f15948ed27aebbf&pid=1-s2.0-S0012365X24003807-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Jacobian of a graph and graph automorphisms","authors":"István Estélyi ,&nbsp;Ján Karabáš ,&nbsp;Alexander Mednykh ,&nbsp;Roman Nedela","doi":"10.1016/j.disc.2024.114259","DOIUrl":"10.1016/j.disc.2024.114259","url":null,"abstract":"<div><p>In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph <em>X</em> in the group of symmetries of the Jacobian of <em>X</em>. As a consequence we show that if a 3-edge-connected graph <em>X</em> admits a nonabelian semiregular group of automorphisms, then the Jacobian of <em>X</em> cannot be cyclic. In particular, Cayley graphs of degree at least three arising from nonabelian groups have non-cyclic Jacobians. While the size of the Jacobian of <em>X</em> is well-understood – it is equal to the number of spanning trees of <em>X</em> – the combinatorial interpretation of the rank of Jacobian of a graph is unknown. Our paper presents a contribution in this direction.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114259"},"PeriodicalIF":0.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Adversarial graph burning densities","authors":"Karen Gunderson ,&nbsp;William Kellough ,&nbsp;J.D. Nir ,&nbsp;Hritik Punj","doi":"10.1016/j.disc.2024.114253","DOIUrl":"10.1016/j.disc.2024.114253","url":null,"abstract":"<div><p>Graph burning is a discrete-time process that models the spread of influence in a network. Vertices are either <em>burning</em> or <em>unburned</em>, and in each round, a burning vertex causes all of its neighbours to become burning before a new <em>fire source</em> is chosen to become burning. We introduce a variation of this process that incorporates an adversarial game played on a nested, growing sequence of graphs. Two players, Arsonist and Builder, play in turns: Builder adds a certain number of new unburned vertices and edges incident to these to create a larger graph, then every vertex neighbouring a burning vertex becomes burning, and finally Arsonist ‘burns’ a new fire source. This process repeats forever. Arsonist is said to win if the limiting fraction of burning vertices tends to 1, while Builder is said to win if this fraction is bounded away from 1.</p><p>The central question of this paper is determining if, given that Builder adds <span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> vertices at turn <em>n</em>, either Arsonist or Builder has a winning strategy. In the case that <span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is asymptotically polynomial, we give threshold results for which player has a winning strategy.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114253"},"PeriodicalIF":0.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003844/pdfft?md5=844834b3c93d41373d6c5d8d83ccf0aa&pid=1-s2.0-S0012365X24003844-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the spum and sum-diameter of paths","authors":"Aryan Bora ,&nbsp;Yunseo Choi ,&nbsp;Lucas Tang","doi":"10.1016/j.disc.2024.114257","DOIUrl":"10.1016/j.disc.2024.114257","url":null,"abstract":"<div><p>In a sum graph, the vertices are labeled with distinct positive integers, and two vertices are adjacent if the sum of their labels is equal to the label of another vertex. In 1990, Harary showed that not all graphs <em>G</em> can be labeled as a sum graph but the union of <em>G</em> and at least some <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> isolated vertices can be. The spum of a graph <em>G</em> is defined as the minimum difference between the largest and smallest labels of a sum graph that consists of the union of <em>G</em> and exactly <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> isolated vertices. More recently, Li introduced the sum-diameter of a graph <em>G</em>, which modifies the definition of spum by removing the requirement that the number of isolated vertices must be <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper, we settle conjectures by Singla, Tiwari, and Tripathi and a conjecture by Li by evaluating the spum and the sum-diameter of paths.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114257"},"PeriodicalIF":0.7,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalized Schröder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials","authors":"Mawo Ito","doi":"10.1016/j.disc.2024.114230","DOIUrl":"10.1016/j.disc.2024.114230","url":null,"abstract":"<div><p>Lattice paths called <em>ℓ</em>-Schröder paths are introduced. They are paths on the upper half-plane consisting of <span><math><mi>ℓ</mi><mo>+</mo><mn>2</mn></math></span> types of steps: <span><math><mo>(</mo><mi>i</mi><mo>,</mo><mi>ℓ</mi><mo>−</mo><mi>i</mi><mo>)</mo></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>ℓ</mi></math></span>, and <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. Those paths generalize Schröder paths and some variants, such as <em>m</em>-Schröder paths by Yang and Jiang and Motzkin-Schröder paths by Kim and Stanton. We show that <em>ℓ</em>-Schröder paths arise naturally from a combinatorial interpretation of the moments of generalized Laurent bi-orthogonal polynomials introduced by Wang, Chang, and Yue. We also show that some generating functions of non-intersecting <em>ℓ</em>-Schröder paths can be factorized in closed forms.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114230"},"PeriodicalIF":0.7,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003613/pdfft?md5=3d23fab95f47d9c48cf5e308a2091300&pid=1-s2.0-S0012365X24003613-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lagrangian densities of 4-uniform matchings and degree stability of extremal hypergraphs","authors":"Zilong Yan ,&nbsp;Yuejian Peng","doi":"10.1016/j.disc.2024.114235","DOIUrl":"10.1016/j.disc.2024.114235","url":null,"abstract":"&lt;div&gt;&lt;p&gt;The Lagrangian density of an &lt;em&gt;r&lt;/em&gt;-uniform graph &lt;em&gt;F&lt;/em&gt; is &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;sup&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&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;G&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mtext&gt;-&lt;/mtext&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&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; is the Lagrangian of an &lt;em&gt;r&lt;/em&gt;-uniform graph &lt;em&gt;G&lt;/em&gt;. Hypergraph Lagrangian has been a helpful tool in extremal combinatorics. Let &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; denote the &lt;em&gt;r&lt;/em&gt;-uniform matching with size &lt;em&gt;t&lt;/em&gt;. The well-known Erdős Matching conjecture proposed that the Turán number of &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is &lt;span&gt;&lt;math&gt;&lt;mi&gt;max&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is the complete &lt;em&gt;r&lt;/em&gt;-graph on &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; vertices and &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the &lt;em&gt;r&lt;/em&gt;-graph with vertex set &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and with edge set &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&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;e&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&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;/math&gt;&lt;/span&gt;. Regarding Lagrangian density of hypergraph matchings, Jiang, Peng and Wu &lt;span&gt;&lt;span&gt;[22]&lt;/span&gt;&lt;/span&gt; (Wu &lt;span&gt;&lt;span&gt;[34]&lt;/span&gt;&lt;/span&gt; as well) conjectured that the property similar to Erdős Matching Conjecture holds, precisely, they conjectured that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;(","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114235"},"PeriodicalIF":0.7,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003662/pdfft?md5=9242f6e0ae9a52d7a2d53e03f6ceabb7&pid=1-s2.0-S0012365X24003662-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142163058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"q-ary (1,k)-overlap-free codes with given restrictions","authors":"Xiaomiao Wang ,&nbsp;Yu Fang ,&nbsp;Tao Feng","doi":"10.1016/j.disc.2024.114236","DOIUrl":"10.1016/j.disc.2024.114236","url":null,"abstract":"<div><p>Two words <em>u</em> and <em>v</em> have a <em>t</em>-overlap if the length <em>t</em> prefix of <em>u</em> is equal to the length <em>t</em> suffix of <em>v</em>, or vice versa. A code <span><math><mi>C</mi></math></span> is <em>t</em>-overlap-free if no two words <em>u</em> and <em>v</em> in <span><math><mi>C</mi></math></span> (including <span><math><mi>u</mi><mo>=</mo><mi>v</mi></math></span>) have a <em>t</em>-overlap. A code of length <em>n</em> is said to be <span><math><mo>(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-overlap-free if it is <em>t</em>-overlap-free for all <em>t</em> such that <span><math><mn>1</mn><mo>⩽</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⩽</mo><mi>t</mi><mo>⩽</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⩽</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. A <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-overlap-free code of length <em>n</em> is called non-overlapping, which has applications in DNA-based data storage systems and frame synchronization. In this paper, we initialize the study for codes of length <em>n</em> which are simultaneously <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>)</mo></math></span>-overlap-free and <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-overlap-free, and establish lower and upper bounds for the size of balanced and error-correcting <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>)</mo></math></span>-overlap-free codes.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114236"},"PeriodicalIF":0.7,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003674/pdfft?md5=832dbe1e406e021c0d775dae451bb738&pid=1-s2.0-S0012365X24003674-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Noncommutative symmetric functions and skewing operators","authors":"Byung-Hak Hwang","doi":"10.1016/j.disc.2024.114255","DOIUrl":"10.1016/j.disc.2024.114255","url":null,"abstract":"<div><p>Skewing operators play a central role in the symmetric function theory because of the importance of the product structure of the symmetric function space. The theory of noncommutative symmetric functions is a useful tool for studying expansions of a given symmetric function in terms of various bases. In this paper, we establish a further development of the theory for studying skewing operators. Using this machinery, we are able to easily reproduce the Littlewood–Richardson rule and provide recurrence relations for chromatic quasisymmetric functions, which generalize Harada–Precup's recurrence.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114255"},"PeriodicalIF":0.7,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003868/pdfft?md5=b3b462b75688640dc1e6facb9ced629f&pid=1-s2.0-S0012365X24003868-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Edge-apexing in hereditary classes of graphs","authors":"Jagdeep Singh ,&nbsp;Vaidy Sivaraman","doi":"10.1016/j.disc.2024.114234","DOIUrl":"10.1016/j.disc.2024.114234","url":null,"abstract":"<div><p>A class <span><math><mi>G</mi></math></span> of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>epex</mi></mrow></msup></math></span> the class of graphs that are at most one edge away from being in <span><math><mi>G</mi></math></span>. We note that <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>epex</mi></mrow></msup></math></span> is hereditary and prove that if a hereditary class <span><math><mi>G</mi></math></span> has finitely many forbidden induced subgraphs, then so does <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>epex</mi></mrow></msup></math></span>.</p><p>The hereditary class of cographs consists of all graphs <em>G</em> that can be generated from <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> using complementation and disjoint union. Cographs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. For the class of edge-apex cographs our main result bounds the order of such forbidden induced subgraphs by 8 and finds all of them by computer search.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114234"},"PeriodicalIF":0.7,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003650/pdfft?md5=3a7b1576f400f1b5803871014f7dd340&pid=1-s2.0-S0012365X24003650-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}