The notion of incompatibility system was first proposed by Krivelevich, Lee and Sudakov to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph , an incompatibility system over is a family such that for every , is a family of edge pairs with . Moreover, for an integer , we say is ‐bounded if for every vertex and its incident edge , there are at most pairs in containing . Krivelevich, Lee and Sudakov proved that there is an universal constant such that for every Dirac graph and every ‐bounded incompatibility system over , there exists a Hamilton cycle where every pair of adjacent edges of satisfies for . This resolves a conjecture posed by Häggkvist in 1988 and such a Hamilton cycle is called compatible (with respect to ). We study high powers of Hamilton cycles in this context and show that for every and , there exists a constant such that for sufficiently large and every ‐bounded incompatibility system over an ‐vertex graph with , there exists a compatible th power of a Hamilton cycle in . Moreover, we give a ‐bounded construction which has minimum degree and contains no compatible th power of a Hamilton cycle.
{"title":"Compatible powers of Hamilton cycles in dense graphs","authors":"Xiaohan Cheng, Jie Hu, Donglei Yang","doi":"10.1002/jgt.23178","DOIUrl":"https://doi.org/10.1002/jgt.23178","url":null,"abstract":"The notion of incompatibility system was first proposed by Krivelevich, Lee and Sudakov to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph , an <jats:italic>incompatibility system</jats:italic> over is a family such that for every , is a family of edge pairs with . Moreover, for an integer , we say is ‐<jats:italic>bounded</jats:italic> if for every vertex and its incident edge , there are at most pairs in containing . Krivelevich, Lee and Sudakov proved that there is an universal constant such that for every Dirac graph and every ‐bounded incompatibility system over , there exists a Hamilton cycle where every pair of adjacent edges of satisfies for . This resolves a conjecture posed by Häggkvist in 1988 and such a Hamilton cycle is called <jats:italic>compatible</jats:italic> (with respect to ). We study high powers of Hamilton cycles in this context and show that for every and , there exists a constant such that for sufficiently large and every ‐bounded incompatibility system over an ‐vertex graph with , there exists a compatible th power of a Hamilton cycle in . Moreover, we give a ‐bounded construction which has minimum degree and contains no compatible th power of a Hamilton cycle.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142263967","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}
Let be a simple graph and let be two integers with . We prove that for every if and only if has a ‐factor, where denotes the number of isolated vertices of and is a special family of trees. Furthermore, we characterize the trees in in terms of their bipartition.
{"title":"Fractional factors and component factors in graphs with isolated toughness smaller than 1","authors":"Isaak H. Wolf","doi":"10.1002/jgt.23179","DOIUrl":"https://doi.org/10.1002/jgt.23179","url":null,"abstract":"Let be a simple graph and let be two integers with . We prove that for every if and only if has a ‐factor, where denotes the number of isolated vertices of and is a special family of trees. Furthermore, we characterize the trees in in terms of their bipartition.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142263968","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}
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 , where 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 .
{"title":"Edge‐transitive cubic graphs of twice square‐free order","authors":"Gui Xian Liu, Zai Ping Lu","doi":"10.1002/jgt.23168","DOIUrl":"https://doi.org/10.1002/jgt.23168","url":null,"abstract":"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 , where 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 .","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-17","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}
For a graph , we define a small automorphism as one that maps some vertex into its neighbour. We investigate the edge colourings of that break every small automorphism of . 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 yields a total colouring which breaks all the small automorphisms of . These results are sharp, and they match the known bounds for the nonlist variant.
{"title":"Breaking small automorphisms by list colourings","authors":"Jakub Kwaśny, Marcin Stawiski","doi":"10.1002/jgt.23181","DOIUrl":"https://doi.org/10.1002/jgt.23181","url":null,"abstract":"For a graph , we define a small automorphism as one that maps some vertex into its neighbour. We investigate the edge colourings of that break every small automorphism of . 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 yields a total colouring which breaks all the small automorphisms of . These results are sharp, and they match the known bounds for the nonlist variant.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-17","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}
A 3‐connected graph is a brick if has a perfect matching, for each pair of vertices of . A brick is minimal if ceases to be a brick for every edge . Norine and Thomas proved that each minimal brick contains at least three vertices of degree three and made a stronger conjecture: there exists such that every minimal brick has at least 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 vertices contains more than vertices of degree at most four.
{"title":"On a Norine–Thomas conjecture concerning minimal bricks","authors":"Xing Feng","doi":"10.1002/jgt.23175","DOIUrl":"https://doi.org/10.1002/jgt.23175","url":null,"abstract":"A 3‐connected graph is a <jats:italic>brick</jats:italic> if has a perfect matching, for each pair of vertices of . A brick is <jats:italic>minimal</jats:italic> if ceases to be a brick for every edge . Norine and Thomas proved that each minimal brick contains at least three vertices of degree three and made a stronger conjecture: there exists such that every minimal brick has at least 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 vertices contains more than vertices of degree at most four.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-11","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
Given a finite family of graphs, we say that a graph is “‐free” if does not contain any graph in as a subgraph. We abbreviate ‐free to just “‐free” when . A vertex‐colored graph is called “rainbow” if no two vertices of have the same color. Given an integer and a finite family of graphs , let denote the smallest integer such that any properly vertex‐colored ‐free graph having contains an induced rainbow path on vertices. Scott and Seymour showed that exists for every complete graph . A conjecture of N. R. Aravind states that . The upper bound on that can be obtained using the methods of Scott and Seymour setting are, however, super‐exponential. Gyárfás and Sárközy showed that . For , we show that and therefore, . This significantly improves Gyárfás and Sárközy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that , where . Moreover, in each case, our results imply the existence of at least distinct induced rainbow paths on vertices. Along the way, we obtain some new results on an oriented variant of the Gyárfás–Sumner conjecture. For , let denote the orientations of in which one vertex has out‐degree or in‐degree . We show that every ‐free oriented graph having a chromatic number at least and every bikernel‐perfect oriented graph with girth having a chromatic number at least contains every oriented tree on at most vertices as an induced subgraph.
{"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":"https://doi.org/10.1002/jgt.23171","url":null,"abstract":"Given a finite family of graphs, we say that a graph is “‐free” if does not contain any graph in as a subgraph. We abbreviate ‐free to just “‐free” when . A vertex‐colored graph is called “rainbow” if no two vertices of have the same color. Given an integer and a finite family of graphs , let denote the smallest integer such that any properly vertex‐colored ‐free graph having contains an induced rainbow path on vertices. Scott and Seymour showed that exists for every complete graph . A conjecture of N. R. Aravind states that . The upper bound on that can be obtained using the methods of Scott and Seymour setting are, however, super‐exponential. Gyárfás and Sárközy showed that . For , we show that and therefore, . This significantly improves Gyárfás and Sárközy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that , where . Moreover, in each case, our results imply the existence of at least distinct induced rainbow paths on vertices. Along the way, we obtain some new results on an oriented variant of the Gyárfás–Sumner conjecture. For , let denote the orientations of in which one vertex has out‐degree or in‐degree . We show that every ‐free oriented graph having a chromatic number at least and every bikernel‐perfect oriented graph with girth having a chromatic number at least contains every oriented tree on at most vertices as an induced subgraph.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-11","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}
An edge of a matching covered graph is removable if 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 is a brick if it is free of nontrivial tight cuts. Carvalho, Lucchesi and Murty proved that every brick other than and has at least removable edges. A brick is near‐bipartite if it has a pair of edges such that is a bipartite matching covered graph. In this paper, we show that in a near‐bipartite brick with at least six vertices, every vertex of , except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, has at least removable edges. Moreover, all graphs attaining this lower bound are characterized.
{"title":"Removable edges in near‐bipartite bricks","authors":"Yipei Zhang, Fuliang Lu, Xiumei Wang, Jinjiang Yuan","doi":"10.1002/jgt.23173","DOIUrl":"https://doi.org/10.1002/jgt.23173","url":null,"abstract":"An edge of a matching covered graph is <jats:italic>removable</jats:italic> if 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 is a <jats:italic>brick</jats:italic> if it is free of nontrivial tight cuts. Carvalho, Lucchesi and Murty proved that every brick other than and has at least removable edges. A brick is <jats:italic>near‐bipartite</jats:italic> if it has a pair of edges such that is a bipartite matching covered graph. In this paper, we show that in a near‐bipartite brick with at least six vertices, every vertex of , except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, has at least removable edges. Moreover, all graphs attaining this lower bound are characterized.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","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}
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.
{"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":null,"pages":null},"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}
Let be any wheel graph and the class of all countable graphs not containing as a minor. We show that there exists a graph in which contains every graph in as an induced subgraph.
{"title":"Universal graphs with forbidden wheel minors","authors":"Thilo Krill","doi":"10.1002/jgt.23174","DOIUrl":"https://doi.org/10.1002/jgt.23174","url":null,"abstract":"Let be any wheel graph and the class of all countable graphs not containing as a minor. We show that there exists a graph in which contains every graph in as an induced subgraph.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A graph is said to be uniquely hamiltonian 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 has a Cayley graph of degree that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of are outerplanar.
{"title":"On vertex‐transitive graphs with a unique hamiltonian cycle","authors":"Babak Miraftab, Dave Witte Morris","doi":"10.1002/jgt.23166","DOIUrl":"https://doi.org/10.1002/jgt.23166","url":null,"abstract":"A graph is said to be <jats:italic>uniquely hamiltonian</jats:italic> 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 has a Cayley graph of degree that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of are outerplanar.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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