Discrete Mathematics最新文献

{"title":"Every subcubic graph is packing (1,1,2,2,3)-colorable","authors":"Xujun Liu ,&nbsp;Xin Zhang ,&nbsp;Yanting Zhang","doi":"10.1016/j.disc.2025.114610","DOIUrl":"10.1016/j.disc.2025.114610","url":null,"abstract":"<div><div>For a sequence <span><math><mi>S</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> of non-decreasing integers, a packing <em>S</em>-coloring of a graph <em>G</em> is a partition of its vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> into <span><math><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>k</mi></mrow></msub></math></span> such that for every pair of distinct vertices <span><math><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, where <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span>, the distance between <em>u</em> and <em>v</em> is at least <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mn>1</mn></math></span>. The packing chromatic number, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of a graph <em>G</em> is the smallest integer <em>k</em> such that <em>G</em> has a packing <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>)</mo></math></span>-coloring. Gastineau and Togni asked an open question “Is it true that the 1-subdivision (<span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>) of any subcubic graph <em>G</em> has packing chromatic number at most 5?” and later Brešar, Klavžar, Rall, and Wash conjectured that it is true.</div><div>In this paper, we prove that every subcubic graph has a packing <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>-coloring and it is sharp due to the existence of subcubic graphs that are not packing <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-colorable. As a corollary of our result, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mn>6</mn></math></span> for every subcubic graph <em>G</em>, improving a previous bound (8) due to Balogh, Kostochka, and Liu in 2019, and we are now just one step away from fully solving the conjecture.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114610"},"PeriodicalIF":0.7,"publicationDate":"2025-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144170356","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 strong odd colorings of graphs","authors":"Yair Caro ,&nbsp;Mirko Petruševski ,&nbsp;Riste Škrekovski ,&nbsp;Zsolt Tuza","doi":"10.1016/j.disc.2025.114601","DOIUrl":"10.1016/j.disc.2025.114601","url":null,"abstract":"<div><div>A strong odd coloring of a simple graph <em>G</em> is a proper coloring of the vertices of <em>G</em> such that for every vertex <em>v</em> and every color <em>c</em>, either <em>c</em> is used an odd number of times in the open neighborhood <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span> or no neighbor of <em>v</em> is colored by <em>c</em>. The smallest integer <em>k</em> for which <em>G</em> admits a strong odd coloring with <em>k</em> colors is the strong odd chromatic number, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>so</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. These coloring notion and graph parameter were recently defined in Kwon and Park (<span><span>arXiv:2401.11653</span><svg><path></path></svg></span>). We answer a question raised by the originators concerning the existence of a constant bound for the strong odd chromatic number of all planar graphs. We also consider strong odd colorings of trees, unicyclic graphs, claw-free graphs, and graph products.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114601"},"PeriodicalIF":0.7,"publicationDate":"2025-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144170355","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":"A lower bound for the energy of graphs in terms of the vertex cover number","authors":"S. Akbari ,&nbsp;S. Küçükçifçi ,&nbsp;H. Saveh ,&nbsp;E.Ş. Yazıcı","doi":"10.1016/j.disc.2025.114582","DOIUrl":"10.1016/j.disc.2025.114582","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The energy of the graph &lt;em&gt;G&lt;/em&gt;, denoted by &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&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 sum of the absolute values of its eigenvalues. Wang and Ma proved that if &lt;em&gt;G&lt;/em&gt; has &lt;em&gt;c&lt;/em&gt; odd cycles, then &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&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;mn&gt;2&lt;/mn&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&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;c&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;β&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 vertex cover number of &lt;em&gt;G&lt;/em&gt;. In this paper we strengthen this result by showing that if &lt;em&gt;G&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; have &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; numbers of induced odd cycles, respectively, then &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&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;mn&gt;2&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&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;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and we conjecture that for every graph &lt;em&gt;G&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&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;mn&gt;2&lt;/mn&gt;&lt;mi&gt;β&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;. We prove the conjecture for some families of graphs, namely, bipartite graphs, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-free regular graphs, perfect graphs, and for all graphs with &lt;span&gt;&lt;math&gt;&lt;mi&gt;β&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;mfrac&gt;&lt;mrow&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;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;. It is shown that for every graph &lt;em&gt;G&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&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;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; is the complement of &lt;em&gt;G&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denote the largest and the smallest eigenvalues of the adjacency ","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114582"},"PeriodicalIF":0.7,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144170357","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":"Some interlacing sequences related to the Eulerian and derangement polynomials","authors":"Xue Yan,&nbsp;Lily Li Liu","doi":"10.1016/j.disc.2025.114598","DOIUrl":"10.1016/j.disc.2025.114598","url":null,"abstract":"<div><div>The Eulerian polynomials and derangement polynomials arise often in combinatorics, algebra and geometry. It is well known that the Eulerian polynomials and derangement polynomials form generalized Sturm sequences, respectively. In this paper, we give new sufficient conditions for the interlacing property of recurrence sequences of polynomials. As applications, we show some interesting interlacing sequences among the Eulerian polynomials, the derangement polynomials and those generalized polynomials for colored permutations.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114598"},"PeriodicalIF":0.7,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144147540","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":"Turán problems for star-path forests in hypergraphs","authors":"Junpeng Zhou ,&nbsp;Xiying Yuan","doi":"10.1016/j.disc.2025.114592","DOIUrl":"10.1016/j.disc.2025.114592","url":null,"abstract":"<div><div>An <em>r</em>-uniform hypergraph (<em>r</em>-graph for short) is linear if any two edges intersect at most one vertex. Let <span><math><mi>F</mi></math></span> be a given family of <em>r</em>-graphs. An <em>r</em>-graph <em>H</em> is called <span><math><mi>F</mi></math></span>-free if <em>H</em> does not contain any member of <span><math><mi>F</mi></math></span> as a subgraph. The Turán number of <span><math><mi>F</mi></math></span> is the maximum number of edges in any <span><math><mi>F</mi></math></span>-free <em>r</em>-graph on <em>n</em> vertices, and the linear Turán number of <span><math><mi>F</mi></math></span> is defined as the Turán number of <span><math><mi>F</mi></math></span> in linear host hypergraphs. An <em>r</em>-uniform linear path <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> of length <em>ℓ</em> is an <em>r</em>-graph with edges <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> such that <span><math><mo>|</mo><mi>V</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>∩</mo><mi>V</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span> if <span><math><mo>|</mo><mi>i</mi><mo>−</mo><mi>j</mi><mo>|</mo><mo>=</mo><mn>1</mn></math></span>, and <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>∩</mo><mi>V</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> for <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span> otherwise. Gyárfás et al. (2022) <span><span>[9]</span></span> obtained an upper bound for the linear Turán number of <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math></span>. In this paper, an upper bound for the linear Turán number of <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> is obtained, which generalizes the known result of <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> to any <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>. Furthermore, some results for the linear Turán number and Turán number of several linear star-path forests are obtained.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114592"},"PeriodicalIF":0.7,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144137779","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":"Forest cuts in sparse graphs","authors":"Vsevolod Chernyshev ,&nbsp;Johannes Rauch ,&nbsp;Dieter Rautenbach","doi":"10.1016/j.disc.2025.114594","DOIUrl":"10.1016/j.disc.2025.114594","url":null,"abstract":"<div><div>We consider the conjecture that every graph <em>G</em> of order <em>n</em> with less than <span><math><mn>3</mn><mi>n</mi><mo>−</mo><mn>6</mn></math></span> edges has a vertex cut that induces a forest. Maximal planar graphs do not have such vertex cuts and show that the density condition would be best possible. We verify the conjecture for planar graphs and show that every graph <em>G</em> of order <em>n</em> with less than <span><math><mfrac><mrow><mn>11</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mi>n</mi><mo>−</mo><mfrac><mrow><mn>18</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> edges has a vertex cut that induces a forest.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114594"},"PeriodicalIF":0.7,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144137780","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 rainbow numbers of paths in maximal bipartite planar graphs","authors":"Lei Ren,&nbsp;Yongxin Lan,&nbsp;Changqing Xu","doi":"10.1016/j.disc.2025.114596","DOIUrl":"10.1016/j.disc.2025.114596","url":null,"abstract":"<div><div>Given two graphs <em>G</em> and <em>T</em>, the rainbow number of <em>T</em> in <em>G</em>, denoted by <span><math><mi>r</mi><mi>b</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span>, is the minimum positive integer <em>t</em> such that, if <em>G</em> contains a copy of <em>T</em>, then every <em>t</em>-edge-coloring of <em>G</em> contains a rainbow copy of <em>T</em>. Given a family of graphs <span><math><mi>G</mi></math></span> and a graph <em>T</em>, if every graph in <span><math><mi>G</mi></math></span> contains a copy of <em>T</em>, then the rainbow number of <em>T</em> in <span><math><mi>G</mi></math></span>, denoted by <span><math><mi>r</mi><mi>b</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span>, is defined as <span><math><mi>max</mi><mo>⁡</mo><mo>{</mo><mi>r</mi><mi>b</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>|</mo><mi>G</mi><mo>∈</mo><mi>G</mi><mo>}</mo></math></span>. Given a graph <em>T</em>, let <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span> denote the family of all maximal bipartite planar graphs on <em>n</em> vertices that contain a copy of <em>T</em>. In this paper, we study the rainbow numbers of paths in maximal bipartite planar graphs, we get the exact value of <span><math><mi>r</mi><mi>b</mi><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mi>ℓ</mi></math></span> and <span><math><mi>ℓ</mi><mo>≠</mo><mn>8</mn></math></span>, and the lower bound of <span><math><mi>r</mi><mi>b</mi><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114596"},"PeriodicalIF":0.7,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134929","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":"Pattern avoidance in revised ascent sequences","authors":"Robin D.P. Zhou","doi":"10.1016/j.disc.2025.114608","DOIUrl":"10.1016/j.disc.2025.114608","url":null,"abstract":"<div><div>Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only if it serves as an ascent bottom. We construct a bijection between ascent sequences and revised ascent sequences by adapting the classic hat map, which transforms ascent sequences into modified ascent sequences. Additionally, we investigate revised ascent sequences that avoid a single pattern, leading to a wealth of enumerative results. Our main techniques include the use of bijections, generating trees, generating functions, and the kernel method.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114608"},"PeriodicalIF":0.7,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134931","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":"Enumerating several statistics of r-colored Dyck paths with no dd-steps having the same colors","authors":"Yidong Sun,&nbsp;Jinyi Wang,&nbsp;Xinyu Wang","doi":"10.1016/j.disc.2025.114597","DOIUrl":"10.1016/j.disc.2025.114597","url":null,"abstract":"<div><div>An <em>r</em>-colored Dyck path is a Dyck path with all <strong>d</strong>-steps having one of <em>r</em> colors in <span><math><mo>[</mo><mi>r</mi><mo>]</mo><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>r</mi><mo>}</mo></math></span>. In this paper, we consider several statistics on the set <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msubsup></math></span> of <em>r</em>-colored Dyck paths of length 2<em>n</em> with no two consecutive <strong>d</strong>-steps having the same colors. Precisely, the paper studies the statistics “number of points” at level <em>ℓ</em>, “number of <strong>u</strong>-steps” at level <span><math><mi>ℓ</mi><mo>+</mo><mn>1</mn></math></span>, “number of peaks” at level <span><math><mi>ℓ</mi><mo>+</mo><mn>1</mn></math></span> and “number of <strong>udu</strong>-steps” on the set <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msubsup></math></span>. The counting formulas of the first three statistics are established by Riordan arrays related to <span><math><mi>S</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>x</mi><mo>)</mo></math></span>, the weighted generating function of <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-Schröder paths. By a useful and surprising relations satisfied by <span><math><mi>S</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>x</mi><mo>)</mo></math></span>, several identities related to these counting formulas are also described.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114597"},"PeriodicalIF":0.7,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134930","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":"Distance-based (and path-based) covering problems for graphs of given cyclomatic number","authors":"Dibyayan Chakraborty ,&nbsp;Florent Foucaud ,&nbsp;Anni Hakanen","doi":"10.1016/j.disc.2025.114595","DOIUrl":"10.1016/j.disc.2025.114595","url":null,"abstract":"<div><div>We study a large family of graph covering problems, whose definitions rely on distances, for graphs of bounded cyclomatic number (that is, the minimum number of edges that need to be removed from the graph to destroy all cycles). These problems include (but are not restricted to) three families of problems: (i) variants of metric dimension, where one wants to choose a small set <em>S</em> of vertices of the graph such that every vertex is uniquely determined by its ordered vector of distances to the vertices of <em>S</em>; (ii) variants of geodetic sets, where one wants to select a small set <em>S</em> of vertices such that any vertex lies on some shortest path between two vertices of <em>S</em>; (iii) variants of path covers, where one wants to select a small set of paths such that every vertex or edge belongs to one of the paths. We generalize and/or improve previous results in the area which show that the optimal values for these problems can be upper-bounded by a linear function of the cyclomatic number and the degree 1-vertices of the graph. To this end, we develop and enhance a technique recently introduced in (Lu et al., 2022 <span><span>[53]</span></span>) and give near-optimal bounds in several cases. This solves (in some cases fully, in some cases partially) some conjectures and open questions from the literature. The method, based on breadth-first search, is of algorithmic nature and thus, all the constructions can be computed in linear time. Our results also imply an algorithmic consequence for the computation of the <em>optimal</em> solutions: for some of the problems, they can be computed in polynomial time for graphs of bounded cyclomatic number.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114595"},"PeriodicalIF":0.7,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116592","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}