{"title":"Edge-transitive cubic graphs of twice square-free order","authors":"Gui Xian Liu, Zai Ping Lu","doi":"10.1002/jgt.23168","DOIUrl":"10.1002/jgt.23168","url":null,"abstract":"<p>A graph is edge-transitive if its automorphism group acts transitively on the edge set. This paper presents a complete classification for connected edge-transitive cubic graphs of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $2n$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> is even and square-free. In particular, it is shown that such a graph is either symmetric or isomorphic to one of the following graphs: a semisymmetric graph of order 420, a semisymmetric graph of order 29,260, and five families of semisymmetric graphs constructed from the simple group <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>PSL</mtext>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>p</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{bf{text{PSL}}}}_{2}(p)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"173-204"},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142263787","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}
Ervin Győri, Addisu Paulos, Nika Salia, Casey Tompkins, Oscar Zamora
{"title":"The maximum number of pentagons in a planar graph","authors":"Ervin Győri, Addisu Paulos, Nika Salia, Casey Tompkins, Oscar Zamora","doi":"10.1002/jgt.23172","DOIUrl":"https://doi.org/10.1002/jgt.23172","url":null,"abstract":"<p>In 1979, Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex planar graph. They precisely determined the maximum number of triangles and four-cycles and presented a conjecture for the maximum number of pentagons. In this work, we confirm their conjecture. Even more, we characterize the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex, planar graphs with the maximum number of pentagons.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"229-256"},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868691","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":"Breaking small automorphisms by list colourings","authors":"Jakub Kwaśny, Marcin Stawiski","doi":"10.1002/jgt.23181","DOIUrl":"10.1002/jgt.23181","url":null,"abstract":"<p>For a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, we define a small automorphism as one that maps some vertex into its neighbour. We investigate the edge colourings of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> that break every small automorphism of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. We show that such a colouring can be chosen from any set of lists of length 3. In addition, we show that any set of lists of length 2 on both edges and vertices of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> yields a total colouring which breaks all the small automorphisms of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. These results are sharp, and they match the known bounds for the nonlist variant.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"288-292"},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142263792","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 a Norine–Thomas conjecture concerning minimal bricks","authors":"Xing Feng","doi":"10.1002/jgt.23175","DOIUrl":"10.1002/jgt.23175","url":null,"abstract":"<p>A 3-connected graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a <i>brick</i> if <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>S</mi>\u0000 </mrow></math> has a perfect matching, for each pair <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> of vertices of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>. A brick <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is <i>minimal</i> if <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>e</mi>\u0000 </mrow></math> ceases to be a brick for every edge <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>E</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math>. Norine and Thomas proved that each minimal brick contains at least three vertices of degree three and made a stronger conjecture: there exists <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>α</mi>\u0000 \u0000 <mo>></mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow></math> such that every minimal brick <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> has at least <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>α</mi>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∣</mo>\u0000 </mrow></math> cubic vertices. In this paper, we prove this conjecture holds for all minimal bricks of an average degree no less than 23/5. As its corollary, we show that each minimal brick on <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math> vertices contains more than <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>5</mn>\u0000 </mrow></math> vertices of degree at most four.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"162-172"},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199688","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}
Manu Basavaraju, L. Sunil Chandran, Mathew C. Francis, Karthik Murali
{"title":"Variants of the Gyárfás–Sumner conjecture: Oriented trees and rainbow paths","authors":"Manu Basavaraju, L. Sunil Chandran, Mathew C. Francis, Karthik Murali","doi":"10.1002/jgt.23171","DOIUrl":"10.1002/jgt.23171","url":null,"abstract":"<p>Given a finite family <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math> of graphs, we say that a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is “<span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math>-free” if <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> does not contain any graph in <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math> as a subgraph. We abbreviate <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math>-free to just “<span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow></math>-free” when <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mi>F</mi>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow></math>. A vertex-colored graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math> is called “rainbow” if no two vertices of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math> have the same color. Given an integer <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>s</mi>\u0000 </mrow></math> and a finite family of graphs <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math>, let <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>s</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ℱ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> denote the smallest integer such that any properly vertex-colored <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math>-free graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> having <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>χ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mi>ℓ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"136-161"},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199689","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":"Corrigendum: The diameter of AT-free graphs","authors":"Guillaume Ducoffe","doi":"10.1002/jgt.23170","DOIUrl":"10.1002/jgt.23170","url":null,"abstract":"<p>This corrigendum corrects an error found in the proof of correctness of the algorithm by [Ducoffe, JGT, 2022, 99(4), pp. 594–614], Theorem 6. An erroneous result from Deogun and Kratsch was used in the original proof. There are no changes in the algorithm itself.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"848-850"},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23170","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199690","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":"Removable edges in near-bipartite bricks","authors":"Yipei Zhang, Fuliang Lu, Xiumei Wang, Jinjiang Yuan","doi":"10.1002/jgt.23173","DOIUrl":"10.1002/jgt.23173","url":null,"abstract":"<p>An edge <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow></math> of a matching covered graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is <i>removable</i> if <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>e</mi>\u0000 </mrow></math> is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. A nonbipartite matching covered graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a <i>brick</i> if it is free of nontrivial tight cuts. Carvalho, Lucchesi and Murty proved that every brick other than <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 </mrow></math> and <span></span><math>\u0000 \u0000 <mrow>\u0000 <mover>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>6</mn>\u0000 </msub>\u0000 \u0000 <mo>¯</mo>\u0000 </mover>\u0000 </mrow></math> has at least <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math> removable edges. A brick <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is <i>near-bipartite</i> if it has a pair of edges <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow></math> such that <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"113-135"},"PeriodicalIF":0.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199691","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":"Universal graphs with forbidden wheel minors","authors":"Thilo Krill","doi":"10.1002/jgt.23174","DOIUrl":"10.1002/jgt.23174","url":null,"abstract":"<p>Let <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>W</mi>\u0000 </mrow></math> be any wheel graph and <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> the class of all countable graphs not containing <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>W</mi>\u0000 </mrow></math> as a minor. We show that there exists a graph in <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> which contains every graph in <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> as an induced subgraph.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"100-112"},"PeriodicalIF":0.9,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23174","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199692","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 vertex-transitive graphs with a unique hamiltonian cycle","authors":"Babak Miraftab, Dave Witte Morris","doi":"10.1002/jgt.23166","DOIUrl":"10.1002/jgt.23166","url":null,"abstract":"<p>A graph is said to be <i>uniquely hamiltonian</i> if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow></math> has a Cayley graph of degree <span></span><math>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math> that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow></math> are outerplanar.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"65-99"},"PeriodicalIF":0.9,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23166","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199693","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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