Gunnar Brinkmann, Kenta Noguchi, Heidi Van den Camp
{"title":"Face Sizes and the Connectivity of the Dual","authors":"Gunnar Brinkmann, Kenta Noguchi, Heidi Van den Camp","doi":"10.1002/jgt.23277","DOIUrl":"https://doi.org/10.1002/jgt.23277","url":null,"abstract":"<p>For each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, we prove tight lower bounds on face sizes that must be present to allow 1- or 2-cuts in simple duals of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-connected maps. Using these bounds, we determine the smallest genus on which a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-connected map can have a simple dual with a 2-cut and give lower and some upper bounds for the smallest genus on which a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-connected map can have a simple dual with a 1-cut.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 4","pages":"379-391"},"PeriodicalIF":1.0,"publicationDate":"2025-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23277","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145272966","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 Minimum Degree of Minimally \u0000 \u0000 \u0000 \u0000 t\u0000 \u0000 \u0000 -Tough, Claw-Free Graphs","authors":"Hui Ma, Xiaomin Hu, Weihua Yang","doi":"10.1002/jgt.23278","DOIUrl":"https://doi.org/10.1002/jgt.23278","url":null,"abstract":"<div>\u0000 \u0000 <p>A graph is called minimally <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-tough if the toughness of the graph is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> but the removal of any edge decreases the toughness. Katona and Varga conjectured that every minimally <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-tough graph has a vertex of degree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>⌈</mo>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>t</mi>\u0000 </mrow>\u0000 \u0000 <mo>⌉</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Matthews and Sumner proved that the toughness of any claw-free graph is always equal to half its connectivity, which implies that the toughness of a claw-free graph is always an integer or half of an integer. Katona et al. proved that this conjecture holds for minimally <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-tough, claw-free graphs if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mfenced>\u0000 <mrow>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mfenced>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, and we proved before that this is also true if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mfrac>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 3","pages":"366-373"},"PeriodicalIF":1.0,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145022285","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}