Journal of Graph Theory最新文献

{"title":"A Construction of a 3/2-Tough Plane Triangulation With No 2-Factor","authors":"Songling Shan","doi":"10.1002/jgt.23209","DOIUrl":"https://doi.org/10.1002/jgt.23209","url":null,"abstract":"<div>\u0000 \u0000 <p>In 1956, Tutte proved the celebrated theorem that every 4-connected planar graph is Hamiltonian. This result implies that every more than <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics>\u0000 </mrow>\u0000 <annotation> $frac{3}{2}$</annotation>\u0000 </semantics></math>-tough planar graph on at least three vertices is Hamiltonian and so has a 2-factor. Owens in 1999 constructed non-Hamiltonian maximal planar graphs of toughness arbitrarily close to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics>\u0000 </mrow>\u0000 <annotation> $frac{3}{2}$</annotation>\u0000 </semantics></math> and asked whether there exists a maximal non-Hamiltonian planar graph of toughness exactly <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics>\u0000 </mrow>\u0000 <annotation> $frac{3}{2}$</annotation>\u0000 </semantics></math>. In fact, the graphs Owens constructed do not even contain a 2-factor. Thus the toughness of exactly <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics>\u0000 </mrow>\u0000 <annotation> $frac{3}{2}$</annotation>\u0000 </semantics></math> is the only case left in asking the existence of 2-factors in tough planar graphs. This question was also asked by Bauer, Broersma, and Schmeichel in a survey. In this paper, we close this gap by cons","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"5-18"},"PeriodicalIF":0.9,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612531","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":"Equitable List Coloring of Planar Graphs With Given Maximum Degree","authors":"H. A. Kierstead, Alexandr Kostochka, Zimu Xiang","doi":"10.1002/jgt.23203","DOIUrl":"https://doi.org/10.1002/jgt.23203","url":null,"abstract":"<p>If <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math> is a list assignment of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math> colors to each vertex of an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, then an <i>equitable</i> <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-<i>coloring</i> of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a proper coloring of vertices of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> from their lists such that no color is used more than <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⌈</mo>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>/</mo>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <mo>⌉</mo>\u0000 </mrow>\u0000 <annotation> $lceil n/rrceil $</annotation>\u0000 </semantics></math> times. A graph is <i>equitably</i> <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>-<i>choosable</i> if it has an equitable <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-coloring for every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>-list assignment <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"832-838"},"PeriodicalIF":0.9,"publicationDate":"2024-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23203","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143456080","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 the Pre- and Post-Positional Semi-Random Graph Processes","authors":"Pu Gao, Hidde Koerts","doi":"10.1002/jgt.23202","DOIUrl":"https://doi.org/10.1002/jgt.23202","url":null,"abstract":"<p>We study the semi-random graph process, and a variant process recently suggested by Nick Wormald. We show that these two processes are asymptotically equally fast in constructing a semi-random graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> that has property <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math>, for the following examples of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math>: (1) <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math> is the set of graphs containing a fixed <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math>-degenerate subgraph, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $dge 1$</annotation>\u0000 </semantics></math> is fixed and (2) <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math> is the set of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-connected graphs, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $kge 1$</annotation>\u0000 </semantics></math> is fixed. In particular, our result of the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-connectedness above settles the open case <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"819-831"},"PeriodicalIF":0.9,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23202","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455949","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 a Question of Erdős and Nešetřil About Minimal Cuts in a Graph","authors":"Domagoj Bradač","doi":"10.1002/jgt.23207","DOIUrl":"https://doi.org/10.1002/jgt.23207","url":null,"abstract":"<p>Answering a question of Erdős and Nešetřil, we show that the maximum number of inclusion-wise minimal vertex cuts in a graph on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> vertices is at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1.889</mn>\u0000 \u0000 <msup>\u0000 <mn>9</mn>\u0000 \u0000 <mi>n</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> $1.889{9}^{n}$</annotation>\u0000 </semantics></math> for large enough <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"817-818"},"PeriodicalIF":0.9,"publicationDate":"2024-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23207","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455745","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":"Hypergraph Anti-Ramsey Theorems","authors":"Xizhi Liu,&nbsp;Jialei Song","doi":"10.1002/jgt.23204","DOIUrl":"https://doi.org/10.1002/jgt.23204","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;The anti-Ramsey number &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;ar&lt;/mtext&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;\u0000 $text{ar}(n,F)$\u0000&lt;/annotation&gt;\u0000 &lt;/semantics&gt;\u0000 &lt;/math&gt; of an &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;\u0000 $r$\u0000&lt;/annotation&gt;\u0000 &lt;/semantics&gt;\u0000 &lt;/math&gt;-graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;\u0000 $F$\u0000&lt;/annotation&gt;\u0000 &lt;/semantics&gt;\u0000 &lt;/math&gt; is the minimum number of colors needed to color the complete &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;\u0000 $n$\u0000&lt;/annotation&gt;\u0000 &lt;/semantics&gt;\u0000 &lt;/math&gt;-vertex &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;\u0000 $r$\u0000&lt;/annotation&gt;\u0000 &lt;/semantics&gt;\u0000 &lt;/math&gt;-graph to ensure the existence of a rainbow copy of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;\u0000 $F$\u0000&lt;/annotation&gt;\u0000 &lt;/semantics&gt;\u0000 &lt;/math&gt;. We establish a removal-type result for the anti-Ramsey problem of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;\u0000 $F$\u0000&lt;/annotation&gt;\u0000 &lt;/semantics&gt;\u0000 &lt;/math&gt; when &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;\u0000 $F$\u0000&lt;/annotation&gt;\u0000 &lt;/semantics&gt;\u0000 &lt;/math&gt; is the expansion of a hypergraph with a smaller uniformity. We present two applications of this result. First, we refine the general bound &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;ar&lt;/mtext&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"808-816"},"PeriodicalIF":0.9,"publicationDate":"2024-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455744","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":"Small Planar Hypohamiltonian Graphs","authors":"Cheng-Chen Tsai","doi":"10.1002/jgt.23205","DOIUrl":"https://doi.org/10.1002/jgt.23205","url":null,"abstract":"<div>\u0000 \u0000 <p>A graph is hypohamiltonian if it is non-hamiltonian, but the deletion of every single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 40 vertices, a result due to Jooyandeh, McKay, Östergård, Pettersson, and Zamfirescu. That result is here improved upon by two planar hypohamiltonian graphs on 34 vertices. We exploited a special subgraph contained in two graphs of Jooyandeh et al., and modified it to construct the two 34-vertex graphs and six planar hypohamiltonian graphs on 37 vertices. Each of the 34-vertex graphs has 26 cubic vertices, improving upon the result of Jooyandeh et al. that planar hypohamiltonian graphs have 30 cubic vertices. We use the 34-vertex graphs to construct hypohamiltonian graphs of order 34 with crossing number 1, improving the best-known bound of 36 due to Wiener. Whether there exists a planar hypohamiltonian graph on 41 vertices was an open question. We settled this question by applying an operation introduced by Thomassen to the 37-vertex graphs to obtain several planar hypohamiltonian graphs on 41 vertices. The 25 planar hypohamiltonian graphs on 40 vertices of Jooyandeh et al. have no nontrivial automorphisms. The result is here improved upon by six planar hypohamiltonian graphs on 40 vertices with nontrivial automorphisms.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"799-807"},"PeriodicalIF":0.9,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455664","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":"Tight Upper Bound on the Clique Size in the Square of 2-Degenerate Graphs","authors":"Seog-Jin Kim,&nbsp;Xiaopan Lian","doi":"10.1002/jgt.23201","DOIUrl":"https://doi.org/10.1002/jgt.23201","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;The &lt;i&gt;square&lt;/i&gt; of a graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, denoted &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${G}^{2}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, has the same vertex set as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and has an edge between two vertices if the distance between them in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is at most 2. In general, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;≤&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;msup&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;≤&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 \u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 \u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{Delta }}(G)+1le chi ({G}^{2})le {rm{Delta }}{(G)}^{2}+1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for every graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Charpentier (2014) asked whether &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;m","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"781-798"},"PeriodicalIF":0.9,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455925","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 maximum number of odd cycles in a planar graph","authors":"Emily Heath,&nbsp;Ryan R. Martin,&nbsp;Chris Wells","doi":"10.1002/jgt.23197","DOIUrl":"https://doi.org/10.1002/jgt.23197","url":null,"abstract":"<p>How many copies of a fixed odd cycle, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${C}_{2m+1}$</annotation>\u0000 </semantics></math>, can a planar graph contain? We answer this question asymptotically for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $min {2,3,4}$</annotation>\u0000 </semantics></math> and prove a bound which is tight up to a factor of 3/2 for all other values of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>. This extends the prior results of Cox and Martin and of Lv, Győri, He, Salia, Tompkins, and Zhu on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>μ</mi>\u0000 </mrow>\u0000 <annotation> $mu $</annotation>\u0000 </semantics></math> on the edges of some clique maximizes the probability that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> edges sampled independently from <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>μ</mi>\u0000 </mrow>\u0000 <annotation> $mu $</annotation>\u0000 </semantics></math> form either a cycle or a path?</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"745-780"},"PeriodicalIF":0.9,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23197","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143456010","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":"Odd chromatic number of graph classes","authors":"Rémy Belmonte,&nbsp;Ararat Harutyunyan,&nbsp;Noleen Köhler,&nbsp;Nikolaos Melissinos","doi":"10.1002/jgt.23200","DOIUrl":"https://doi.org/10.1002/jgt.23200","url":null,"abstract":"&lt;p&gt;A graph is called &lt;i&gt;odd&lt;/i&gt; (respectively, &lt;i&gt;even&lt;/i&gt;) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an &lt;i&gt;odd colouring&lt;/i&gt; of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Scott proved that a connected graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-odd colourable if it can be partitioned into at most &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; odd induced subgraphs. The &lt;i&gt;odd chromatic number of&lt;/i&gt; &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, denoted by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 \u0000 &lt;mtext&gt;odd&lt;/mtext&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${chi }_{text{odd}}(G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, is the minimum integer &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for which &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-odd colourable. We initiate the systematic study of odd colouring and odd chromatic number of graph classes. We fir","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"722-744"},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23200","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455839","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 effect of symmetry-preserving operations on 3-connectivity","authors":"Heidi Van den Camp","doi":"10.1002/jgt.23196","DOIUrl":"https://doi.org/10.1002/jgt.23196","url":null,"abstract":"<p>In 2017, Brinkmann, Goetschalckx and Schein introduced a very general way of describing operations on embedded graphs that preserve all orientation-preserving symmetries of the graph. This description includes all well-known operations such as Dual, Truncation and Ambo. As these operations are applied locally, they are called local orientation-preserving symmetry-preserving operations (lopsp-operations). In this text, we will use the general description of these operations to determine their effect on 3-connectivity. Recently it was proved that all lopsp-operations preserve 3-connectivity of graphs that have face-width at least three. We present a simple condition that characterises exactly which lopsp-operations preserve 3-connectivity for all embedded graphs, even for those with face-width less than three.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"672-704"},"PeriodicalIF":0.9,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455666","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}