Discrete Mathematics最新文献

{"title":"Ideally connected cographs and chordal graphs","authors":"Richter Jordaan","doi":"10.1016/j.disc.2025.114819","DOIUrl":"10.1016/j.disc.2025.114819","url":null,"abstract":"<div><div>For distinct vertices <span><math><mi>u</mi><mo>,</mo><mi>v</mi></math></span> in a graph <em>G</em>, let <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span> denote the maximum number of internally disjoint <em>u</em>-<em>v</em> paths in <em>G</em>. Then, <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><msub><mrow><mtext>deg</mtext></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><msub><mrow><mtext>deg</mtext></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>}</mo></math></span>. If equality is attained for every pair of vertices in <em>G</em>, then <em>G</em> is called <em>ideally connected</em>. In this paper, we characterize the ideally connected graphs in two well-known graph classes: the cographs and the chordal graphs. We show that the ideally connected cographs are precisely the <span><math><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free cographs, and the ideally connected chordal graphs are precisely the threshold graphs, the graphs that can be constructed from the single-vertex graph by repeatedly adding either an isolated vertex or a dominating vertex.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114819"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219190","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":"Divisible design graphs with selfloops","authors":"Anwita Bhowmik ,&nbsp;Bart De Bruyn ,&nbsp;Sergey Goryainov","doi":"10.1016/j.disc.2025.114824","DOIUrl":"10.1016/j.disc.2025.114824","url":null,"abstract":"<div><div>We develop a basic theory for divisible design graphs with possible selfloops (LDDG's), and describe two infinite families of such graphs, some members of which are also classical examples of divisible design graphs without loops (DDG's). Among the described theoretical results is a discussion of the spectrum, a classification of all examples satisfying certain parameter restrictions or having at most three eigenvalues, a discussion of the structure of the improper and the disconnected examples, and a procedure called dual Seidel switching which allows to construct new examples of LDDG's from others.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114824"},"PeriodicalIF":0.7,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227085","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 strong Bordeaux Conjecture","authors":"Xiangwen Li ,&nbsp;Lin Niu,&nbsp;Fangyu Tian","doi":"10.1016/j.disc.2025.114818","DOIUrl":"10.1016/j.disc.2025.114818","url":null,"abstract":"<div><div>Steinberg Conjecture (1976) states that every planar graph without 4-cycles and 5-cycles is 3-colorable, and the strong Bordeaux Conjecture (2003) says that every planar graph without 5-cycles and adjacent 3-cycles is 3-colorable. In 2017, such both conjectures are disproved by Cohen–Addad et al. In the view of improper coloring, one naturally asks whether every planar graph without 4-cycles and 5-cycles is <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-colorable and whether every planar graph without 5-cycles and adjacent 3-cycles is <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-colorable. In this paper, we prove that every planar graph without 5-cycles and adjacent 3-cycles is <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-colorable, which improves the early results of Li et al. (2020) <span><span>[11]</span></span>, Chen et al. (2016) <span><span>[3]</span></span> and Liu et al. (2015) <span><span>[12]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114818"},"PeriodicalIF":0.7,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227088","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 minimally t-tough graphs with t ≤ 1","authors":"Shiyu Cao,&nbsp;Jing Chen,&nbsp;Wei Zheng","doi":"10.1016/j.disc.2025.114814","DOIUrl":"10.1016/j.disc.2025.114814","url":null,"abstract":"<div><div>Let <em>t</em> be a positive real number. If the toughness of <em>G</em> is <em>t</em> and the deletion of any edge from <em>G</em> decreases its toughness, then <em>G</em> is a minimally <em>t</em>-tough graph. The generalized Kriesell's conjecture states that there exists a vertex of degree <span><math><mo>⌈</mo><mn>2</mn><mi>t</mi><mo>⌉</mo></math></span> in each minimally <em>t</em>-tough graph. The conjecture is recently disproved in general, but in this paper we prove it for some families of graphs. In this paper, we mainly discuss minimally <em>t</em>-tough and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msub></math></span>-free graphs with <span><math><mi>r</mi><mo>≥</mo><mn>4</mn></math></span> and the structures of minimally <em>t</em>-tough graphs with <span><math><mi>t</mi><mo>≤</mo><mn>1</mn></math></span>. We prove that the conjecture stated above holds for minimally <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>-tough and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>4</mn></mrow></msub></math></span>-free graphs, and minimally <em>t</em>-tough graphs with a simplicial vertex (a vertex where all its neighbors are also adjacent to each other) and <span><math><mi>t</mi><mo>≤</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114814"},"PeriodicalIF":0.7,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157629","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":"Bridge indices of spatial graphs and diagram colorings","authors":"Sarah Blackwell ,&nbsp;Puttipong Pongtanapaisan ,&nbsp;Hanh Vo","doi":"10.1016/j.disc.2025.114813","DOIUrl":"10.1016/j.disc.2025.114813","url":null,"abstract":"<div><div>We extend the Wirtinger number of links, an invariant originally defined by Blair, Kjuchukova, Velazquez, and Villanueva in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove that the Wirtinger number equals the bridge index of spatial graphs, and we implement an algorithm in Python which gives a more efficient way to estimate upper bounds of bridge indices. Combined with lower bounds from diagram colorings by elements from certain algebraic structures and clasping techniques, we obtain exact bridge indices for a large family of almost unknotted spatial graphs. We also show that for every possible negative Euler characteristic, there exist almost unknotted graphs of arbitrarily large bridge index.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114813"},"PeriodicalIF":0.7,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158888","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":"Distinguishing symmetric digraphs by proper arc-colourings of type I","authors":"Rafał Kalinowski,&nbsp;Monika Pilśniak,&nbsp;Magdalena Prorok","doi":"10.1016/j.disc.2025.114816","DOIUrl":"10.1016/j.disc.2025.114816","url":null,"abstract":"<div><div>A symmetric digraph <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>↔</mo></mrow></mover></math></span> is obtained from a simple graph <em>G</em> by replacing each edge <em>uv</em> with a pair of opposite arcs <span><math><mover><mrow><mi>u</mi><mi>v</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>, <span><math><mover><mrow><mi>v</mi><mi>u</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>. An arc-colouring <em>c</em> of a digraph <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>↔</mo></mrow></mover></math></span> is distinguishing if the only automorphism of <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>↔</mo></mrow></mover></math></span> preserving the colouring <em>c</em> is the identity. Behzad introduced the proper arc-colouring of type I as an arc-colouring such that any two consecutive arcs <span><math><mover><mrow><mi>u</mi><mi>v</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>, <span><math><mover><mrow><mi>v</mi><mi>w</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span> have distinct colours. We establish an optimal upper bound <span><math><mo>⌈</mo><mn>2</mn><msqrt><mrow><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msqrt><mo>⌉</mo></math></span> for the least number of colours in a distinguishing proper colouring of type I of a connected symmetric digraph <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>↔</mo></mrow></mover></math></span>. Furthermore, we prove that the same upper bound <span><math><mo>⌈</mo><mn>2</mn><msqrt><mrow><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msqrt><mo>⌉</mo></math></span> is optimal for another type of proper colouring of <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>↔</mo></mrow></mover></math></span>, when only monochromatic 2-paths are forbidden.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114816"},"PeriodicalIF":0.7,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158822","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 connection between the chromatic numbers of a hypergraph and its 1-intersection graph","authors":"Zoltán L. Blázsik ,&nbsp;Nathan W. Lemons","doi":"10.1016/j.disc.2025.114810","DOIUrl":"10.1016/j.disc.2025.114810","url":null,"abstract":"<div><div>A well known problem from an excellent book of Lovász states that any hypergraph with the property that no pair of hyperedges intersect in exactly one vertex can be properly 2-colored. Motivated by this as well as recent works of Keszegh and of Gyárfás et al. we study the 1-intersection graph of a hypergraph. The 1-intersection graph encodes those pairs of hyperedges in a hypergraph that intersect in exactly one vertex. We prove for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>}</mo></math></span> that all hypergraphs whose 1-intersection graph is <em>k</em>-partite can be properly <em>k</em>-colored.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114810"},"PeriodicalIF":0.7,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157630","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 flip graph on planar layouts of a planar tanglegram is almost a hypercube","authors":"Kevin Liu","doi":"10.1016/j.disc.2025.114815","DOIUrl":"10.1016/j.disc.2025.114815","url":null,"abstract":"<div><div>Given a planar layout of a planar tanglegram, it is known that all other planar layouts can be obtained using paired flips at leaf-matched pairs of vertices. Consequently, for any planar tanglegram <span><math><mi>T</mi></math></span>, the paired flip operation generates a connected flip graph <span><math><mi>G</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> on the set of planar layouts of <span><math><mi>T</mi></math></span>. We introduce a special subset of leaf-matched pairs that we call <em>essential</em> and show that restricting to paired flips at these pairs generates a hypercube graph on the planar layouts of <span><math><mi>T</mi></math></span>. One consequence of this result is a method of efficiently counting the number of distinct planar tanglegram layouts of a planar tanglegram.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114815"},"PeriodicalIF":0.7,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158889","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":"Resistance, oddness and colouring defect of snarks","authors":"Imran Allie","doi":"10.1016/j.disc.2025.114804","DOIUrl":"10.1016/j.disc.2025.114804","url":null,"abstract":"<div><div>Let <em>G</em> be a bridgeless cubic graph. The <em>resistance</em> of <em>G</em>, denoted <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the minimum number of edges which can be removed from <em>G</em> in order to render 3-edge-colourability. The <em>oddness</em> of <em>G</em>, denoted <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the minimum number of odd components in any 2-factor of <em>G</em>. The <em>colouring defect</em> of <em>G</em> (or simply, the <em>defect</em> of <em>G</em>), denoted <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the minimum number of edges not contained in any set of three perfect matchings of <em>G</em>. These three parameters are regarded as measurements of uncolourability of snarks, partly because all of these parameters equal zero if and only if <em>G</em> is 3-edge-colourable. It is also known that <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and that <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> <span><span>[5]</span></span>, <span><span>[6]</span></span>. We have shown that the ratio of oddness to resistance can be arbitrarily large for non-trivial snarks <span><span>[1]</span></span>. It has also been shown that the ratio of the defect to oddness can be arbitrarily large for non-trivial snarks, although this result was only shown for graphs with oddness equal to 2 <span><span>[7]</span></span>. In the same paper, the question was posed whether there exists non-trivial snarks for given resistance <em>r</em> or given oddness <em>ω</em>, and arbitrarily large defect. In this paper, we prove a stronger result: For any positive integers <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span>, even <span><math><mi>ω</mi><mo>≥</mo><mi>r</mi></math></span>, and <span><math><mi>d</mi><mo>≥</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>ω</mi></math></span>, there exists a non-trivial snark <em>G</em> with <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>r</mi></math></span>, <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>ω</mi></math></span> and <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>d</mi></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114804"},"PeriodicalIF":0.7,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119470","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":"Bidirected graphs, integral quadratic forms and some Diophantine equations","authors":"Jesús Arturo Jiménez González ,&nbsp;Andrzej Mróz","doi":"10.1016/j.disc.2025.114811","DOIUrl":"10.1016/j.disc.2025.114811","url":null,"abstract":"<div><div>Bidirected graphs are multigraphs where every edge has an independent direction at each end. In the paper, with an arbitrary bidirected graph we associate a non-negative integral quadratic form (called the incidence form of the graph), and determine all forms that appear in this way in two main results: first, among non-negative connected unit forms, precisely those of Dynkin type <span><math><mi>A</mi></math></span> or <span><math><mi>D</mi></math></span> are incidence forms; second, we give simple conditions on the coefficients of a non-negative connected non-unitary form to be an incidence form. We say that those non-unitary forms have Dynkin type <span><math><mi>C</mi></math></span>, and justify such nomenclature by generalizing known classifications and properties of non-negative integral quadratic forms of Dynkin types <span><math><mi>A</mi></math></span> and <span><math><mi>D</mi></math></span> to the introduced type <span><math><mi>C</mi></math></span>. We also show that the graphical framework of an incidence form is an useful tool to visualize its arithmetical properties, to prove new facts and to perform efficient computations for integral quadratic forms and related problems in number theory, algebra and graph theory. For instance, in a third main result we relate the walks of a bidirected graph with the <span><math><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></span>-roots of the associated incidence form (and to the classical root systems in the positive case). Moreover, we prove the universality property for a large class of integral quadratic forms, provide computational methods to find solutions or to characterize the finiteness of the sets of solutions of various related Diophantine equations, show a variant of Whitney's theorem on line graphs using switching classes, and apply our techniques to give a conceptual and constructive proof of the non-negativity (and possible Dynkin types) of the Euler quadratic forms of a class of finite-dimensional gentle algebras.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114811"},"PeriodicalIF":0.7,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119468","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}