Journal of Combinatorial Theory Series B最新文献_第7页

The automorphism group of a complementary prism 互补棱柱的自形群

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-08-02 DOI: 10.1016/j.jctb.2024.07.004

Marko Orel

{"title":"The automorphism group of a complementary prism","authors":"Marko Orel","doi":"10.1016/j.jctb.2024.07.004","DOIUrl":"10.1016/j.jctb.2024.07.004","url":null,"abstract":"<div>Given a finite simple graph Γ on n vertices its complementary prism is the graph <math><mi>Γ</mi><mover><mrow><mi>Γ</mi></mrow><mrow><mo>¯</mo></mrow></mover></math> that is obtained from Γ and its complement <math><mover><mrow><mi>Γ</mi></mrow><mrow><mo>¯</mo></mrow></mover></math> by adding a perfect matching where each its edge connects two copies of the same vertex in Γ and <math><mover><mrow><mi>Γ</mi></mrow><mrow><mo>¯</mo></mrow></mover></math>. It generalizes the Petersen graph, which is obtained if Γ is the pentagon. The automorphism group of <math><mi>Γ</mi><mover><mrow><mi>Γ</mi></mrow><mrow><mo>¯</mo></mrow></mover></math> is described for an arbitrary graph Γ. In particular, it is shown that the ratio between the cardinalities of the automorphism groups of <math><mi>Γ</mi><mover><mrow><mi>Γ</mi></mrow><mrow><mo>¯</mo></mrow></mover></math> and Γ can attain only the values 1, 2, 4, and 12. It is shown that <math><mi>Γ</mi><mover><mrow><mi>Γ</mi></mrow><mrow><mo>¯</mo></mrow></mover></math> is vertex-transitive if and only if Γ is vertex-transitive and self-complementary. Moreover, the complementary prism is not a Cayley graph whenever <math><mi>n</mi><mo>></mo><mn>1</mn></math>.</div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"169 ","pages":"Pages 406-429"},"PeriodicalIF":1.2,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000637/pdfft?md5=a7e845989152de594006704697688b0c&pid=1-s2.0-S0095895624000637-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141952091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

H-factors in graphs with small independence number 小独立数图形中的 H 因子

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-07-31 DOI: 10.1016/j.jctb.2024.07.005

Ming Chen , Jie Han , Guanghui Wang , Donglei Yang

引用次数: 0

A weak box-perfect graph theorem 弱箱完全图定理

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-07-30 DOI: 10.1016/j.jctb.2024.07.006

Patrick Chervet , Roland Grappe

{"title":"A weak box-perfect graph theorem","authors":"Patrick Chervet , Roland Grappe","doi":"10.1016/j.jctb.2024.07.006","DOIUrl":"10.1016/j.jctb.2024.07.006","url":null,"abstract":"<div>A graph G is called perfect if <math><mi>ω</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math> for every induced subgraph H of G, where <math><mi>ω</mi><mo>(</mo><mi>H</mi><mo>)</mo></math> is the clique number of H and <math><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math> its chromatic number. The Weak Perfect Graph Theorem of Lovász states that a graph G is perfect if and only if its complement <math><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></math> is perfect. This does not hold for box-perfect graphs, which are the perfect graphs whose stable set polytope is box-totally dual integral.We prove that both G and <math><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></math> are box-perfect if and only if <math><msup><mrow><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>+</mo></mrow></msup></math> is box-perfect, where <math><msup><mrow><mi>G</mi></mrow><mrow><mo>+</mo></mrow></msup></math> is obtained by adding a universal vertex to G. Consequently, <math><msup><mrow><mi>G</mi></mrow><mrow><mo>+</mo></mrow></msup></math> is box-perfect if and only if <math><msup><mrow><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>+</mo></mrow></msup></math> is box-perfect. As a corollary, we characterize when the complete join of two graphs is box-perfect.</div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"169 ","pages":"Pages 367-372"},"PeriodicalIF":1.2,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000650/pdfft?md5=353ef0de641409c4b03042060f5fe02a&pid=1-s2.0-S0095895624000650-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141952089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Boundary rigidity of CAT(0) cube complexes CAT(0) 立方体复合物的边界刚度

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-07-22 DOI: 10.1016/j.jctb.2024.07.003

Jérémie Chalopin, Victor Chepoi

引用次数: 0

Fractional coloring with local demands and applications to degree-sequence bounds on the independence number 具有局部要求的分数着色及其在独立数度序界限中的应用

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-07-22 DOI: 10.1016/j.jctb.2024.07.002

Tom Kelly , Luke Postle

{"title":"Fractional coloring with local demands and applications to degree-sequence bounds on the independence number","authors":"Tom Kelly , Luke Postle","doi":"10.1016/j.jctb.2024.07.002","DOIUrl":"10.1016/j.jctb.2024.07.002","url":null,"abstract":"<div>In a fractional coloring, vertices of a graph are assigned measurable subsets of the real line and adjacent vertices receive disjoint subsets; the fractional chromatic number of a graph is at most k if it has a fractional coloring in which each vertex receives a subset of <math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math> of measure at least <math><mn>1</mn><mo>/</mo><mi>k</mi></math>. We introduce and develop the theory of “fractional colorings with local demands” wherein each vertex “demands” a certain amount of color that is determined by local parameters such as its degree or the clique number of its neighborhood. This framework provides the natural setting in which to generalize degree-sequence type bounds on the independence number. Indeed, by Linear Programming Duality, all of the problems we study have an equivalent formulation as a problem concerning weighted independence numbers, and they often imply new bounds on the independence number.Our results and conjectures are inspired by many of the most classical results and important open problems concerning the independence number and the chromatic number, often simultaneously. We conjecture a local strengthening of both Shearer's bound on the independence number of triangle-free graphs and the fractional relaxation of Molloy's recent bound on their chromatic number, as well as a longstanding problem of Ajtai et al. on the independence number of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math>-free graphs and the fractional relaxations of Reed's <math><mi>ω</mi><mo>,</mo><mi>Δ</mi><mo>,</mo><mi>χ</mi></math> Conjecture and the Total Coloring Conjecture. We prove an approximate version of the first two, and we prove “local demands” versions of Vizing's Theorem and of some χ-boundedness results.</div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"169 ","pages":"Pages 298-337"},"PeriodicalIF":1.2,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An oriented discrepancy version of Dirac's theorem 狄拉克定理的定向差异版本

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-07-22 DOI: 10.1016/j.jctb.2024.06.008

Andrea Freschi, Allan Lo

{"title":"An oriented discrepancy version of Dirac's theorem","authors":"Andrea Freschi, Allan Lo","doi":"10.1016/j.jctb.2024.06.008","DOIUrl":"10.1016/j.jctb.2024.06.008","url":null,"abstract":"<div>The study of graph discrepancy problems, initiated by Erdős in the 1960s, has received renewed attention in recent years. In general, given a 2-edge-coloured graph G, one is interested in embedding a copy of a graph H in G with large discrepancy (i.e. the copy of H contains significantly more than half of its edges in one colour).Motivated by this line of research, Gishboliner, Krivelevich and Michaeli considered an oriented version of graph discrepancy for Hamilton cycles. In particular, they conjectured the following generalisation of Dirac's theorem: if G is an oriented graph on <math><mi>n</mi><mo>≥</mo><mn>3</mn></math> vertices with <math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>/</mo><mn>2</mn></math>, then G contains a Hamilton cycle with at least <math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> edges pointing forwards. In this paper, we present a full resolution to this conjecture.</div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"169 ","pages":"Pages 338-351"},"PeriodicalIF":1.2,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000571/pdfft?md5=c4578c27cd9e214edf177c194b1972bb&pid=1-s2.0-S0095895624000571-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Erdős-Gyárfás function f(n,4,5)=56n+o(n) — So Gyárfás was right Erdős-Gyárfás 函数 f(n,4,5)=56n+o(n) - 所以 Gyárfás 是对的

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-07-22 DOI: 10.1016/j.jctb.2024.07.001

Patrick Bennett , Ryan Cushman , Andrzej Dudek , Paweł Prałat

引用次数: 0

Improved bounds for the zeros of the chromatic polynomial via Whitney's Broken Circuit Theorem 通过惠特尼断路定理改进色度多项式的零点界限

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-07-16 DOI: 10.1016/j.jctb.2024.06.005

Matthew Jenssen , Viresh Patel , Guus Regts

{"title":"Improved bounds for the zeros of the chromatic polynomial via Whitney's Broken Circuit Theorem","authors":"Matthew Jenssen , Viresh Patel , Guus Regts","doi":"10.1016/j.jctb.2024.06.005","DOIUrl":"10.1016/j.jctb.2024.06.005","url":null,"abstract":"<div>We prove that for any graph G of maximum degree at most Δ, the zeros of its chromatic polynomial <math><msub><mrow><mi>χ</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> (in <math><mi>C</mi></math>) lie inside the disc of radius 5.94Δ centered at 0. This improves on the previously best known bound of approximately 6.91Δ.We also obtain improved bounds for graphs of high girth. We prove that for every g there is a constant <math><msub><mrow><mi>K</mi></mrow><mrow><mi>g</mi></mrow></msub></math> such that for any graph G of maximum degree at most Δ and girth at least g, the zeros of its chromatic polynomial <math><msub><mrow><mi>χ</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> lie inside the disc of radius <math><msub><mrow><mi>K</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>Δ</mi></math> centered at 0, where <math><msub><mrow><mi>K</mi></mrow><mrow><mi>g</mi></mrow></msub></math> is the solution to a certain optimization problem. In particular, <math><msub><mrow><mi>K</mi></mrow><mrow><mi>g</mi></mrow></msub><mo><</mo><mn>5</mn></math> when <math><mi>g</mi><mo>≥</mo><mn>5</mn></math> and <math><msub><mrow><mi>K</mi></mrow><mrow><mi>g</mi></mrow></msub><mo><</mo><mn>4</mn></math> when <math><mi>g</mi><mo>≥</mo><mn>25</mn></math> and <math><msub><mrow><mi>K</mi></mrow><mrow><mi>g</mi></mrow></msub></math> tends to approximately 3.86 as <math><mi>g</mi><mo>→</mo><mo>∞</mo></math>.Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph G to the generating function of so-called broken-circuit-free forests in G. We also establish a zero-free disc for the generating function of all forests in G (aka the partition function of the arboreal gas) which may be of independent interest.</div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"169 ","pages":"Pages 233-252"},"PeriodicalIF":1.2,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S009589562400056X/pdfft?md5=75decf318d359a608bc9f520805078ff&pid=1-s2.0-S009589562400056X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141630474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Linkages and removable paths avoiding vertices 避开顶点的连线和可移动路径

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-07-15 DOI: 10.1016/j.jctb.2024.06.006

Xiying Du, Yanjia Li, Shijie Xie , Xingxing Yu

引用次数: 0

Spectral arbitrariness for trees fails spectacularly 树木的光谱任意性失败得很厉害

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-07-14 DOI: 10.1016/j.jctb.2024.06.007

Shaun M. Fallat , H. Tracy Hall , Rupert H. Levene , Seth A. Meyer , Shahla Nasserasr , Polona Oblak , Helena Šmigoc

引用次数: 0