Discrete Mathematics最新文献

{"title":"Adversarial graph burning densities","authors":"","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":null,"pages":null},"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":"","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":null,"pages":null},"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":"","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":null,"pages":null},"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":"","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":null,"pages":null},"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":"","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":null,"pages":null},"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":"","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":null,"pages":null},"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":"","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":null,"pages":null},"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}

{"title":"A note on clique immersion of strong product graphs","authors":"","doi":"10.1016/j.disc.2024.114237","DOIUrl":"10.1016/j.disc.2024.114237","url":null,"abstract":"<div><p>Let <span><math><mi>G</mi><mo>,</mo><mi>H</mi></math></span> be graphs, and <span><math><mi>G</mi><mo>⁎</mo><mi>H</mi></math></span> represent a specific graph product of <em>G</em> and <em>H</em>. Define <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> as the largest <em>t</em> for which <em>G</em> contains a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>-immersion. Collins, Heenehan, and McDonald posed the question: given <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>t</mi></math></span> and <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>r</mi></math></span>, how large can <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>⁎</mo><mi>H</mi><mo>)</mo></math></span> be? They conjectured <span><math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>⁎</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mi>t</mi><mi>r</mi></math></span> when ⁎ denotes the strong product. In this note, we affirm that the conjecture holds for graphs with certain immersions, in particular when <em>H</em> contains <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> as a subgraph. As a consequence we also get an alternative argument for a result of Guyer and McDonald, showing that the line graphs of constant-multiplicity multigraphs satisfy the conjecture originally proposed by Abu-Khzam and Langston.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003686/pdfft?md5=2175b8b68439085105021d9c5e79d193&pid=1-s2.0-S0012365X24003686-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136787","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":"Covering the edges of a graph with triangles","authors":"","doi":"10.1016/j.disc.2024.114226","DOIUrl":"10.1016/j.disc.2024.114226","url":null,"abstract":"<div><p>In a graph <em>G</em>, let <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mo>△</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denote the minimum size of a set of edges and triangles that cover all edges of <em>G</em>, and let <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the maximum size of an edge set that contains at most one edge from each triangle. Motivated by a question of Erdős, Gallai, and Tuza, we study the relationship between <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mo>△</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and establish a sharp upper bound on <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mo>△</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. We also prove Nordhaus-Gaddum-type inequalities for the considered invariants.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003571/pdfft?md5=fe63daa1972dde10572b653b88b81a86&pid=1-s2.0-S0012365X24003571-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142129957","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":"Degree powers and number of stars in graphs with a forbidden broom","authors":"","doi":"10.1016/j.disc.2024.114232","DOIUrl":"10.1016/j.disc.2024.114232","url":null,"abstract":"<div><p>Given a graph <em>G</em> with degree sequence <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and a positive integer <em>r</em>, let <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>. We denote by <span><math><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> the largest value of <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> among <em>n</em>-vertex <em>F</em>-free graphs <em>G</em>, and by <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></math></span> the largest number of stars <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> in <em>n</em>-vertex <em>F</em>-free graphs. The <em>broom</em> <span><math><mi>B</mi><mo>(</mo><mi>ℓ</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> is the graph obtained from an <em>ℓ</em>-vertex path by adding <em>s</em> new leaves connected to a penultimate vertex <em>v</em> of the path.</p><p>We determine <span><math><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>B</mi><mo>(</mo><mi>ℓ</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>)</mo></math></span> for <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span>, any <span><math><mi>ℓ</mi><mo>,</mo><mi>s</mi></math></span> and sufficiently large <em>n</em>, proving a conjecture of Lan, Liu, Qin and Shi. We also determine <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>B</mi><mo>(</mo><mi>ℓ</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>)</mo></math></span> for <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span>, any <span><math><mi>ℓ</mi><mo>,</mo><mi>s</mi></math></span> and sufficiently large <em>n</em>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003637/pdfft?md5=f6743e7fcba7f41401daef264d1fc9cb&pid=1-s2.0-S0012365X24003637-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142129959","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}