Discrete Mathematics最新文献

{"title":"Turán numbers and switching","authors":"","doi":"10.1016/j.disc.2024.114275","DOIUrl":"10.1016/j.disc.2024.114275","url":null,"abstract":"<div><div>Using a switching operation on tournaments we obtain some new lower bounds on the Turán number of the <em>r</em>-graph on <span><math><mi>r</mi><mo>+</mo><mn>1</mn></math></span> vertices with 3 edges. For <span><math><mi>r</mi><mo>=</mo><mn>4</mn></math></span>, extremal examples were constructed using Paley tournaments in previous work. We show that these examples are unique (in a particular sense) using Fourier analysis.</div><div>A 3-tournament is a ‘higher order’ version of a tournament given by an alternating function on triples of distinct vertices in a vertex set. We show that 3-tournaments also enjoy a switching operation and use this to give a formula for the size of a switching class in terms of level permutations, generalising a result of Babai–Cameron.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426774","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":"The edge coloring of the Cartesian product of signed graphs","authors":"","doi":"10.1016/j.disc.2024.114276","DOIUrl":"10.1016/j.disc.2024.114276","url":null,"abstract":"<div><div>According to Vizing's Theorem, a major question in the area of edge coloring is to determine whether a graph is Class 1 or 2. In 1984, Mohar proved that the Cartesian product <span><math><mi>G</mi><mo>□</mo><mi>H</mi></math></span> is Class 1 if <em>G</em> is Class 1 or both <em>G</em> and <em>H</em> have a perfect matching. Recently, Behr proved that the signed graph version of Vizing's Theorem: a signed graph <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> is either Class 1 or 2. Hence, we want to generalize Mohar's results to signed graphs. In this paper, we prove that <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo><mo>□</mo><mo>(</mo><mi>H</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span> is Class 1 if one of the factors, say <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>, is Class 1 and there exists an edge coloring of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> that satisfies a certain property, which is necessary as shown by an example. Let Δ-matching be a matching which covers every vertex of maximum degree. We also show that if both of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>H</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span> have a Δ-matching and at least one of <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>Δ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> is even, then <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo><mo>□</mo><mo>(</mo><mi>H</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span> is Class 1. This implies that if both of <em>G</em> and <em>H</em> have a Δ-matching, then <span><math><mi>G</mi><mo>□</mo><mi>H</mi></math></span> is Class 1, thereby slightly improving upon Mohar's results.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426773","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":"Linear colouring of binomial random graphs","authors":"","doi":"10.1016/j.disc.2024.114278","DOIUrl":"10.1016/j.disc.2024.114278","url":null,"abstract":"<div><div>We investigate the linear chromatic number <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>lin</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo><mo>)</mo></math></span> of the binomial random graph <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> on <em>n</em> vertices in which each edge appears independently with probability <span><math><mi>p</mi><mo>=</mo><mi>p</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. For a graph <em>G</em>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>lin</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is defined as the smallest <em>k</em> such that <em>G</em> admits a <em>k</em>-colouring with the property that every path <em>P</em> in <em>G</em> receives a colour which appears on only one vertex of <em>P</em>. For dense random graphs (<span><math><mi>n</mi><mi>p</mi><mo>→</mo><mo>∞</mo></math></span> as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>), we show that asymptotically almost surely <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>lin</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo><mo>)</mo><mo>≥</mo><mi>n</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>n</mi><mi>p</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo><mo>)</mo><mo>=</mo><mi>n</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></math></span>. Understanding the order of the linear chromatic number for subcritical random graphs (<span><math><mi>n</mi><mi>p</mi><mo>&lt;</mo><mn>1</mn></math></span>) and critical ones (<span><math><mi>n</mi><mi>p</mi><mo>=</mo><mn>1</mn></math></span>) is relatively easy. However, supercritical sparse random graphs (<span><math><mi>n</mi><mi>p</mi><mo>=</mo><mi>c</mi></math></span> for some constant <span><math><mi>c</mi><mo>&gt;</mo><mn>1</mn></math></span>) remain to be investigated.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426775","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":"The Collatz map analogue in polynomial rings and in completions","authors":"","doi":"10.1016/j.disc.2024.114273","DOIUrl":"10.1016/j.disc.2024.114273","url":null,"abstract":"<div><div>We study an analogue of the Collatz map in the polynomial ring <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, where <em>R</em> is an arbitrary commutative ring. We prove that if <em>R</em> is of positive characteristic, then every polynomial in <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, respectively. We also consider the Collatz map on the ring of formal power series <span><math><mi>R</mi><mo>[</mo><mo>[</mo><mi>x</mi><mo>]</mo><mo>]</mo></math></span> when <em>R</em> is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of 2-adic integers, extending previous results of Lagarias.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416658","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 hat guessing game on cactus graphs and cycles","authors":"","doi":"10.1016/j.disc.2024.114272","DOIUrl":"10.1016/j.disc.2024.114272","url":null,"abstract":"<div><div>We study the hat guessing game on graphs. In this game, a player is placed on each vertex <em>v</em> of a graph <em>G</em> and assigned a colored hat from <span><math><mi>h</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> possible colors. Each player makes a deterministic guess on their hat color based on the colors assigned to the players on neighboring vertices, and the players win if at least one player correctly guesses his assigned color. If there exists a strategy that ensures at least one player guesses correctly for every possible assignment of colors, the game defined by <span><math><mo>〈</mo><mi>G</mi><mo>,</mo><mi>h</mi><mo>〉</mo></math></span> is called winning. The hat guessing number of <em>G</em> is the largest integer <em>q</em> so that if <span><math><mi>h</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mi>q</mi></math></span> for all <span><math><mi>v</mi><mo>∈</mo><mi>G</mi></math></span> then <span><math><mo>〈</mo><mi>G</mi><mo>,</mo><mi>h</mi><mo>〉</mo></math></span> is winning.</div><div>In this note, we determine whether <span><math><mo>〈</mo><mi>G</mi><mo>,</mo><mi>h</mi><mo>〉</mo></math></span> is winning for any <em>h</em> whenever <em>G</em> is a cycle, resolving a conjecture of Kokhas and Latyshev in the affirmative and extending it. We then use this result to determine the hat guessing number of every cactus graph, graphs in which every pair of cycles share at most one vertex.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416654","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 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>&lt;</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}