Discrete Mathematics最新文献

{"title":"On the rainbow planar Turán number of paths","authors":"Ervin Győri ,&nbsp;Ryan R. Martin ,&nbsp;Addisu Paulos ,&nbsp;Casey Tompkins ,&nbsp;Kitti Varga","doi":"10.1016/j.disc.2025.114523","DOIUrl":"10.1016/j.disc.2025.114523","url":null,"abstract":"<div><div>An edge-colored graph is said to contain a rainbow-<em>F</em> if it contains <em>F</em> as a subgraph and every edge of <em>F</em> is a distinct color. The problem of maximizing the number of edges among <em>n</em>-vertex properly edge-colored graphs not containing a rainbow-<em>F</em>, known as the rainbow Turán problem, was initiated by Keevash, Mubayi, Sudakov, and Verstraëte. We investigate a variation of this problem with the additional restriction that the graph is planar and we denote the corresponding extremal number by <span><math><msubsup><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>. In particular, we determine <span><math><msubsup><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> denotes the 5-vertex path.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114523"},"PeriodicalIF":0.7,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799495","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":"On the combinatorial structure and algebraic characterizations of distance-regular digraphs","authors":"Giusy Monzillo ,&nbsp;Safet Penjić","doi":"10.1016/j.disc.2025.114512","DOIUrl":"10.1016/j.disc.2025.114512","url":null,"abstract":"<div><div>Let <span><math><mi>Γ</mi><mo>=</mo><mi>Γ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> denote a simple strongly connected digraph with vertex set <em>X</em>, diameter <em>D</em>, and let <span><math><mo>{</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>A</mi><mo>:</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>}</mo></math></span> denote the set of distance-<em>i</em> matrices of Γ. Let <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>D</mi></mrow></msubsup></math></span> denote a partition of <span><math><mi>X</mi><mo>×</mo><mi>X</mi></math></span>, where <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo>{</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>∈</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>|</mo><msub><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>}</mo></math></span> <span><math><mo>(</mo><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>D</mi><mo>)</mo></math></span>. In the literature, such a digraph Γ is said to be <em>distance-regular</em> if <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>D</mi></mrow></msubsup><mo>)</mo></math></span> is a commutative association scheme. In this paper, we provide a combinatorial definition of a distance-regular digraph in terms of equitable partitions. From this definition, we rediscover all well-known algebraic characterizations of such digraphs, including the above one. We also give several new characterizations, and one of them is the spectral excess theorem for distance-regular digraphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114512"},"PeriodicalIF":0.7,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786053","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":"Synchronicity of descent and excedance enumerators in the alternating subgroup","authors":"Umesh Shankar","doi":"10.1016/j.disc.2025.114521","DOIUrl":"10.1016/j.disc.2025.114521","url":null,"abstract":"<div><div>Generalising the work of Dey <span><span>[2]</span></span>, we define the notion of ultra-synchronicity of sequences of real numbers. Let <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> be the number of even permutations of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span> with <em>k</em> descents, odd permutations with <em>k</em> descents, even permutations with <em>k</em> excedances and odd permutations with <em>k</em> excedances, respectively. We show that the four sequences are ultra-synchronised for all <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>. This proves a strengthening of two conjectures of Dey <span><span>[2]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114521"},"PeriodicalIF":0.7,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786052","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":"The Li-Feng transformation of weighted adjacency matrices for graphs with degree-based edge-weights","authors":"Jing Gao,&nbsp;Xueliang Li,&nbsp;Ning Yang,&nbsp;Ruiling Zheng","doi":"10.1016/j.disc.2025.114520","DOIUrl":"10.1016/j.disc.2025.114520","url":null,"abstract":"<div><div>For a graph <em>G</em>, let <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>v</mi></mrow></msub></math></span> be the degree of a vertex <em>v</em>. Given a symmetric real function <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>, the weight of edge <em>uv</em> in graph <em>G</em> is equal to the value <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>)</mo></math></span>. The degree-based weighted adjacency matrix is defined as <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, in which the <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span>-entry is equal to <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>)</mo></math></span> if <em>uv</em> is an edge of <em>G</em> and 0 otherwise. In this paper, we consider the Li-Feng transformation and show that if a graph <em>G</em> contains two pendant paths on a common vertex, the uniform distribution of pendant paths increases the largest eigenvalue of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, when <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is increasing in <em>x</em> and the length of two pendant paths should be at least 2. We also consider the cycle version of Li-Feng transformation and show that if a graph <em>G</em> contains two pendant cycles on a common vertex, the uniform distribution of pendant cycles decreases the largest eigenvalue of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, when <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>&gt;</mo><mn>2</mn><mi>f</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. The purpose of this paper is to unify the study of the graph operation on the largest eigenvalue for the degree-based weighted adjacency matrix.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114520"},"PeriodicalIF":0.7,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786302","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":"Interlacing property of polynomial sequences related to multinomial coefficients","authors":"Ming-Jian Ding","doi":"10.1016/j.disc.2025.114522","DOIUrl":"10.1016/j.disc.2025.114522","url":null,"abstract":"<div><div>In this paper, we show that several consecutive generating functions of the multinomial coefficients form an interlacing sequence. As applications, we provide a positive response to a question proposed by Fisk regarding the interlacing property for zeros of polynomials, which are generated by the central trinomial (quadrinomial) coefficients.</div><div>Furthermore, we prove that some classical polynomial sequences also possess the interlacing property. These sequences include the (weak) exceedance polynomial sequence on involutions in the symmetric group, Motzkin polynomial sequence, local <em>h</em>-polynomial sequences of the cluster subdivision of Cartan-Killing types <em>A</em>, <em>B</em> and <em>D</em>, Narayana polynomial sequences of types <em>A</em> and <em>B</em>, and others.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114522"},"PeriodicalIF":0.7,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777021","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":"Bipartite binding number, k-factor and spectral radius of bipartite graphs","authors":"Yifang Hao ,&nbsp;Shuchao Li ,&nbsp;Yuantian Yu","doi":"10.1016/j.disc.2025.114511","DOIUrl":"10.1016/j.disc.2025.114511","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The binding number &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&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; of a graph &lt;em&gt;G&lt;/em&gt; is the minimum value of &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; taken over all non-empty subsets &lt;em&gt;S&lt;/em&gt; of &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&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; such that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;mi&gt;V&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;. The bipartite binding number &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&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; of a bipartite graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is defined to be &lt;span&gt;&lt;math&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; if &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;munder&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mrow&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;∅&lt;/mo&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;mi&gt;Y&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;munder&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mrow&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&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;mi&gt;Y&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⊊&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; otherwise. Fan and Lin &lt;span&gt;&lt;span&gt;[9]&lt;/span&gt;&lt;/span&gt; investigated &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&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; from spectral perspectives, and provided tight sufficient conditions in terms of the spectral radius of a graph &lt;em&gt;G&lt;/em&gt; to guarantee &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&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;r&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;em&gt;r&lt;/em&gt; is a positive integer. The study of the existence of &lt;em&gt;k&lt;/em&gt;-factors in graphs is a classic problem in graph theory. Fan and Lin &lt;span&gt;&lt;span&gt;[9]&lt;/span&gt;&lt;/span&gt; also provided the spectral radius conditions for 1-binding graphs to contain a perfect matching and a 2-factor, respectively. In this paper, we consider the bipartite analogues of those results obtained in &lt;span&gt;&lt;span&gt;[9]&lt;/sp","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114511"},"PeriodicalIF":0.7,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777617","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":"Exploring the influence of graph operations on zero forcing sets","authors":"Krishna Menon ,&nbsp;Anurag Singh","doi":"10.1016/j.disc.2025.114516","DOIUrl":"10.1016/j.disc.2025.114516","url":null,"abstract":"<div><div>Zero forcing in graphs is a coloring process where a vertex colored blue can <em>force</em> its unique uncolored neighbor to be colored blue. A zero forcing set is a set of initially blue vertices capable of eventually coloring all vertices of the graph. In this paper, we focus on the numbers <span><math><mi>z</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>i</mi><mo>)</mo></math></span>, which is the number of zero forcing sets of size <em>i</em> of the graph <em>G</em>. These numbers were initially studied by Boyer et al. <span><span>[5]</span></span> where they conjectured that for any graph <em>G</em> on <em>n</em> vertices, <span><math><mi>z</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>i</mi><mo>)</mo><mo>≤</mo><mi>z</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mi>i</mi><mo>)</mo></math></span> for all <span><math><mi>i</mi><mo>≥</mo><mn>1</mn></math></span> where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the path graph on <em>n</em> vertices. The main aim of this paper is to show that several classes of graphs, including outerplanar graphs and threshold graphs, satisfy this conjecture. We do this by studying various graph operations and examining how they affect the number of zero forcing sets.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114516"},"PeriodicalIF":0.7,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777027","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 bipartite graphs with the minimum number of spanning trees","authors":"Shicai Gong,&nbsp;Yue Xu,&nbsp;Peng Zou,&nbsp;Jiaxin Wang","doi":"10.1016/j.disc.2025.114514","DOIUrl":"10.1016/j.disc.2025.114514","url":null,"abstract":"<div><div>The collection of all (simple and connected) bipartite graphs with cyclomatic number <em>ω</em> is denoted by <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>. We use <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>;</mo><mi>b</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span> to denote the graph obtained from the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> by removing <span><math><mi>a</mi><mo>−</mo><mi>c</mi></math></span> edges that are all connected to the same vertex of degree <em>a</em>, here <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span> and <em>c</em> are integers with <span><math><mn>2</mn><mo>≤</mo><mi>c</mi><mo>&lt;</mo><mi>a</mi><mo>≤</mo><mi>b</mi></math></span>. The term <span><math><mi>S</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes the skeleton of the graph <em>G</em>, which is defined as the largest induced subgraph of <em>G</em> that contains no pendant vertices.</div><div>In this paper, we investigate the problem of characterizing the graphs within <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span> that possess the minimum number of spanning trees. We show that the skeleton of each graph with the minimum number of spanning trees in <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span> is either <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span>, where <em>a</em> and <em>b</em> are positive integers with <span><math><mn>2</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>b</mi></math></span> and <span><math><mo>(</mo><mi>a</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>ω</mi></math></span>, or <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>;</mo><mi>b</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span>, where <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span> and <em>c</em> are positive integers satisfying <span><math><mn>2</mn><mo>≤</mo><mi>c</mi><mo>&lt;</mo><mi>a</mi><mo>≤</mo><mi>b</mi></math></span> and <span><math><mi>c</mi><mo>−</mo><mn>1</mn><mo>+</mo><mo>(</mo><mi>a</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>ω</mi></math></span>. In addition, we establish some structural properties by the method of analysis to further reduce those candidate graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114514"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759069","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":"On the maximum number of r-cliques in graphs free of complete r-partite subgraphs","authors":"József Balogh ,&nbsp;Suyun Jiang ,&nbsp;Haoran Luo","doi":"10.1016/j.disc.2025.114508","DOIUrl":"10.1016/j.disc.2025.114508","url":null,"abstract":"<div><div>We estimate the maximum possible number of cliques of size <em>r</em> in an <em>n</em>-vertex graph free of a fixed complete <em>r</em>-partite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></math></span>. By viewing every <em>r</em>-clique as a hyperedge, the upper bound on the Turán number of the complete <em>r</em>-partite hypergraphs gives the upper bound <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn><mo>/</mo><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mo>)</mo></mrow></math></span>. We improve this to <span><math><mi>o</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn><mo>/</mo><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mo>)</mo></mrow></math></span>. The main tool in our proof is the graph removal lemma. We also provide several lower bound constructions.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114508"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761179","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":"The saturation number of C6","authors":"Yongxin Lan ,&nbsp;Yongtang Shi ,&nbsp;Yiqiao Wang ,&nbsp;Junxue Zhang","doi":"10.1016/j.disc.2025.114504","DOIUrl":"10.1016/j.disc.2025.114504","url":null,"abstract":"<div><div>A graph <em>G</em> is called <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-saturated if <em>G</em> is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-free but <span><math><mi>G</mi><mo>+</mo><mi>e</mi></math></span> is not for any <span><math><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span>. The saturation number of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, denoted <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, is the minimum number of edges in a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-saturated graph on <em>n</em> vertices. Finding the exact values of <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> has been one of the most intriguing open problems in extremal graph theory. In this paper, we study the saturation number of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span>. We prove that <span><math><mn>4</mn><mi>n</mi><mo>/</mo><mn>3</mn><mo>−</mo><mn>2</mn><mo>≤</mo><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>)</mo><mo>≤</mo><mo>(</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>3</mn></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>9</mn></math></span>, which significantly improves the existing lower and upper bounds for <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114504"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761183","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}