{"title":"On graphs with maximum difference between game chromatic number and chromatic number","authors":"","doi":"10.1016/j.disc.2024.114271","DOIUrl":"10.1016/j.disc.2024.114271","url":null,"abstract":"<div><div>In the vertex colouring game on a graph <em>G</em>, Maker and Breaker alternately colour vertices of <em>G</em> from a palette of <em>k</em> colours, with no two adjacent vertices allowed the same colour. Maker seeks to colour the whole graph while Breaker seeks to make some vertex impossible to colour. The game chromatic number of <em>G</em>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>g</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the minimum number <em>k</em> of colours for which Maker has a winning strategy for the vertex colouring game.</div><div>Matsumoto proved in 2019 that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>g</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>−</mo><mn>1</mn></math></span>, and conjectured that the only equality cases are some graphs of small order and the Turán graph <span><math><mi>T</mi><mo>(</mo><mn>2</mn><mi>r</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>. We resolve this conjecture in the affirmative by considering a modification of the vertex colouring game wherein Breaker may remove a vertex instead of colouring it.</div><div>Matsumoto further asked whether a similar result could be proved for the vertex marking game, and we provide an example to show that no such nontrivial result can exist.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323608","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":"Stabbing boxes with finitely many axis-parallel lines and flats","authors":"","doi":"10.1016/j.disc.2024.114269","DOIUrl":"10.1016/j.disc.2024.114269","url":null,"abstract":"<div><div>In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> to be pierceable by finitely many axis-parallel <em>k</em>-flats, where <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>d</mi></math></span>. We also consider <em>colorful</em> generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-problem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323609","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":"Transversal coalitions in hypergraphs","authors":"","doi":"10.1016/j.disc.2024.114267","DOIUrl":"10.1016/j.disc.2024.114267","url":null,"abstract":"<div><div>A transversal in a hypergraph <em>H</em> is set of vertices that intersect every edge of <em>H</em>. A transversal coalition in <em>H</em> consists of two disjoint sets of vertices <em>X</em> and <em>Y</em> of <em>H</em>, neither of which is a transversal but whose union <span><math><mi>X</mi><mo>∪</mo><mi>Y</mi></math></span> is a transversal in <em>H</em>. Such sets <em>X</em> and <em>Y</em> are said to form a transversal coalition. A transversal coalition partition in <em>H</em> is a vertex partition <span><math><mi>Ψ</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>}</mo></math></span> such that for all <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>p</mi><mo>]</mo></math></span>, either the set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a singleton set that is a transversal in <em>H</em> or the set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> forms a transversal coalition with another set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> for some <em>j</em>, where <span><math><mi>j</mi><mo>∈</mo><mo>[</mo><mi>p</mi><mo>]</mo><mo>∖</mo><mo>{</mo><mi>i</mi><mo>}</mo></math></span>. The transversal coalition number <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo></math></span> in <em>H</em> equals the maximum order of a transversal coalition partition in <em>H</em>. For <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> a hypergraph <em>H</em> is <em>k</em>-uniform if every edge of <em>H</em> has cardinality <em>k</em>. Among other results, we prove that if <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <em>H</em> is a <em>k</em>-uniform hypergraph, then <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mo>⌊</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⌋</mo><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. Further we show that for every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there exists a <em>k</em>-uniform hypergraph that achieves equality in this upper bound.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314146","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":"Fibonacci and Catalan paths in a wall","authors":"","doi":"10.1016/j.disc.2024.114268","DOIUrl":"10.1016/j.disc.2024.114268","url":null,"abstract":"<div><div>We study the distribution of some statistics (width, number of steps, length, area) defined for paths contained in walls. We present the results by giving generating functions, asymptotic approximations, as well as some closed formulas. We prove algebraically that paths in walls of a given width and ending on the <em>x</em>-axis are enumerated by the Catalan numbers, and we provide a bijection between these paths and Dyck paths. We also find that paths in walls with a given number of steps are enumerated by the Fibonacci numbers. Finally, we give a constructive bijection between the paths in walls of a given length and peakless Motzkin paths of the same length.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314147","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 inclusion chromatic index of a Halin graph","authors":"","doi":"10.1016/j.disc.2024.114266","DOIUrl":"10.1016/j.disc.2024.114266","url":null,"abstract":"<div><div>An inclusion-free edge-coloring of a graph <em>G</em> with <span><math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span> is a proper edge-coloring such that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. The minimum number of colors needed in an inclusion-free edge-coloring of <em>G</em> is called the <span><math><mi>i</mi><mi>n</mi><mi>c</mi><mi>l</mi><mi>u</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span>-<span><math><mi>f</mi><mi>r</mi><mi>e</mi><mi>e</mi></math></span> <span><math><mi>c</mi><mi>h</mi><mi>r</mi><mi>o</mi><mi>m</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>c</mi><mspace></mspace><mi>i</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>x</mi></math></span>, denoted by <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>⊂</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper, we show that for a Halin graph <em>G</em> with maximum degree <span><math><mi>Δ</mi><mo>≥</mo><mn>4</mn></math></span>, if <em>G</em> is isomorphic to a wheel <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>Δ</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> where Δ is odd, then <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>⊂</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>+</mo><mn>2</mn></math></span>, otherwise <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>⊂</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span>. We also show a special cubic Halin graph with <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>⊂</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>5</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312178","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":"Cyclic sieving and dihedral sieving on noncrossing (1,2)-configurations","authors":"","doi":"10.1016/j.disc.2024.114262","DOIUrl":"10.1016/j.disc.2024.114262","url":null,"abstract":"<div><p>Verifying a suspicion of Propp and Reiner concerning the cyclic sieving phenomenon (CSP), M. Thiel introduced a Catalan object called noncrossing <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-configurations (denoted by <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>), which is a class of set partitions of <span><math><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>. More precisely, Thiel proved that, with a natural action of the cyclic group <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the triple <span><math><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>)</mo></math></span> exhibits the CSP, where <span><math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>≔</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mo>[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac><msub><mrow><mo>[</mo><mtable><mtr><mtd><mn>2</mn><mi>n</mi></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> is MacMahon's <em>q</em>-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring <span><math><mi>F</mi><mi>D</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, Jesse Kim found a combinatorial basis for <span><math><mi>F</mi><mi>D</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> indexed by <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In this paper, we continue to study <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and obtain the following results:</p><ul><li><span>(1)</span><span><p>We define a statistic on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> whose generating function is <span><math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, which answers a problem of Thiel.</p></span></li><li><span>(2)</span><span><p>We show that <span><math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> is equivalent to<span><span><span><math><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mi>k</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mtd></mtr><mtr><mtd><mn>2</mn><mi>k</mi><mo>+</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></munder><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>n</mi><mo>−</mo><mn","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271003","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":"Graphs with the minimum spectral radius for given independence number","authors":"","doi":"10.1016/j.disc.2024.114265","DOIUrl":"10.1016/j.disc.2024.114265","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> be the set of connected graphs with order <em>n</em> and independence number <em>α</em>. The graph with the minimum spectral radius among <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> is called the minimizer graph. Stevanović in the classical book [Spectral Radius of Graphs, Academic Press, Amsterdam, 2015] pointed out that determining the minimizer graph in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> appears to be a tough problem. Recently, Lou and Guo (2022) <span><span>[14]</span></span> proved that the minimizer graph in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> must be a tree if <span><math><mi>α</mi><mo>≥</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span>. In this paper, we further give the structural features for the minimizer graph in detail, and then provide a constructing theorem for it. Thus, theoretically we determine the minimizer graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> along with their spectral radius for any given <span><math><mi>α</mi><mo>≥</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span>. As an application, we determine all the minimizer graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> for <span><math><mi>α</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>5</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>6</mn></math></span> along with their spectral radius.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271002","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 note on rainbow stackings of random edge-colorings of hypergraphs","authors":"","doi":"10.1016/j.disc.2024.114261","DOIUrl":"10.1016/j.disc.2024.114261","url":null,"abstract":"<div><p>A rainbow stacking of <em>r</em>-edge-colorings <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> of the complete <em>d</em>-uniform hypergraph on <em>n</em> vertices is a way of superimposing <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> so that no edges of the same color are superimposed on each other. The definition of rainbow stackings of graphs was proposed by Alon, Defant, and Kravitz, and they determined a sharp threshold for <em>r</em> (as a function of <em>m</em> and <em>n</em>) governing the existence and nonexistence of rainbow stackings of random <em>r</em>-edge-colorings <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In this paper, we extend their result to <em>d</em>-uniform hypergraph, obtain a sharp threshold for <em>r</em> controlling the existence and nonexistence of rainbow stackings of random <em>r</em>-edge-colorings <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> of the complete <em>d</em>-uniform hypergraph for <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271000","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":"Group divisible designs with block size 4 and group sizes 4 and 10 and some other 4-GDDs","authors":"","doi":"10.1016/j.disc.2024.114254","DOIUrl":"10.1016/j.disc.2024.114254","url":null,"abstract":"<div><p>In this paper, we consider the existence of group divisible designs (GDDs) with block size 4 and group sizes 4 and 10. We show that a 4-GDD of type <span><math><msup><mrow><mn>4</mn></mrow><mrow><mi>t</mi></mrow></msup><msup><mrow><mn>10</mn></mrow><mrow><mi>s</mi></mrow></msup></math></span> exists when the necessary conditions are satisfied, except possibly for a finite number of cases with <span><math><mn>4</mn><mi>t</mi><mo>+</mo><mn>10</mn><mi>s</mi><mo>≤</mo><mn>178</mn></math></span>. We also give some new examples of 4-GDDs for which the number of points is 51, 54 or some value less than or equal to 50.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003856/pdfft?md5=0cc847d0e63ce2f36a39c3a9c8057cf4&pid=1-s2.0-S0012365X24003856-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271001","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":"Critically 3-frustrated signed graphs","authors":"","doi":"10.1016/j.disc.2024.114258","DOIUrl":"10.1016/j.disc.2024.114258","url":null,"abstract":"<div><p>Extending the notion of maxcut, the study of the frustration index of signed graphs is one of the basic questions in the theory of signed graphs. Recently two of the authors initiated the study of critically frustrated signed graphs. That is a signed graph whose frustration index decreases with the removal of any edge. The main focus of this study is on critical signed graphs which are not edge-disjoint unions of critically frustrated signed graphs (namely indecomposable signed graphs) and which are not built from other critically frustrated signed graphs by subdivision. We conjecture that for any given <em>k</em> there are only finitely many critically <em>k</em>-frustrated signed graphs of this kind.</p><p>Providing support for this conjecture we show that there are only two of such critically 3-frustrated signed graphs where there is no pair of edge-disjoint negative cycles. Similarly, we show that there are exactly ten critically 3-frustrated signed planar graphs that are neither decomposable nor subdivisions of other critically frustrated signed graphs. We present a method for building indecomposable critically frustrated signed graphs based on two given such signed graphs. We also show that the condition of being indecomposable is necessary for our conjecture.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003893/pdfft?md5=71ed29caccfef97965ce5a62c57baeb5&pid=1-s2.0-S0012365X24003893-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271485","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}
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