Journal of Graph Theory最新文献

Counting circuit double covers

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-10-02 DOI: 10.1002/jgt.23187

Radek Hušek, Robert Šámal

{"title":"Counting circuit double covers","authors":"Radek Hušek, Robert Šámal","doi":"10.1002/jgt.23187","DOIUrl":"https://doi.org/10.1002/jgt.23187","url":null,"abstract":"We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${C}_{k}$</annotation>\u0000 </semantics></math> for some <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>) instead of cycles (graphs with all degrees even). We give an almost-exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${2}^{n/2-1}$</annotation>\u0000 </semantics></math> circuit double covers and we show an infinite class of graphs for which this bound is tight.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"374-395"},"PeriodicalIF":0.9,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23187","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868031","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}

引用次数: 0

Non-Hamiltonian Cartesian products of two even dicycles

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-30 DOI: 10.1002/jgt.23185

Kenta Noguchi, Carol T. Zamfirescu

{"title":"Non-Hamiltonian Cartesian products of two even dicycles","authors":"Kenta Noguchi, Carol T. Zamfirescu","doi":"10.1002/jgt.23185","DOIUrl":"https://doi.org/10.1002/jgt.23185","url":null,"abstract":"In this note it is proven that there exist infinitely many positive integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <annotation> $a$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $b$</annotation>\u0000 </semantics></math> such that the Cartesian product of a directed cycle of length <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>a</mi>\u0000 </mrow>\u0000 <annotation> $2a$</annotation>\u0000 </semantics></math> and a directed cycle of length <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $2b$</annotation>\u0000 </semantics></math> is non-Hamiltonian. In particular, the Cartesian product of an 880-dicycle and a 4368-dicycle is non-Hamiltonian. We also prove that there is no such graph on fewer than <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>880</mn>\u0000 \u0000 <mo>⋅</mo>\u0000 \u0000 <mn>4368</mn>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>843</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>840</mn>\u0000 </mrow>\u0000 <annotation> $880cdot 4368=3,843,840$</annotation>\u0000 </semantics></math> vertices, which is rather astonishing.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"368-373"},"PeriodicalIF":0.9,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142869244","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}

引用次数: 0

An extension of Nash-Williams and Tutte's Theorem

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-26 DOI: 10.1002/jgt.23189

Xuqian Fang, Daqing Yang

{"title":"An extension of Nash-Williams and Tutte's Theorem","authors":"Xuqian Fang, Daqing Yang","doi":"10.1002/jgt.23189","DOIUrl":"https://doi.org/10.1002/jgt.23189","url":null,"abstract":"The celebrated Nash-Williams and Tutte's Theorem states that a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> contains <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> edge-disjoint spanning trees if and only if <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>ν</mi>\u0000 \u0000 <mi>f</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${nu }_{f}(G)ge k$</annotation>\u0000 </semantics></math>, where\u0000\u0000 In this paper, we prove that, for integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $kge 0$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $dge 1$</annotation>\u0000 </semantics></math>, if <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>ν</mi>\u0000 \u0000 <mi>f</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>></mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mfrac>\u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"361-367"},"PeriodicalIF":0.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142869216","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}

引用次数: 0

Graph curvature and local discrepancy

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-24 DOI: 10.1002/jgt.23176

Paul Horn, Adam Purcilly, Alex Stevens

{"title":"Graph curvature and local discrepancy","authors":"Paul Horn, Adam Purcilly, Alex Stevens","doi":"10.1002/jgt.23176","DOIUrl":"https://doi.org/10.1002/jgt.23176","url":null,"abstract":"In recent years, discrete notions of curvature have been defined and exploited to understand various geometric properties of graphs; especially regarding heat flow, and spectral properties. In this paper, we study various combinatorial properties implied by satisfying the Bakry–Émery curvature dimension inequality <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 \u0000 <mi>D</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>∞</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>K</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $CD(infty ,K)$</annotation>\u0000 </semantics></math>. In particular we derive a local discrepancy inequality, similar in spirit to the expander mixing lemma from spectral graph theory, which certifies a type of “local pseudo-randomness” of the edge set of the graph, for graphs satisfying a curvature lower bound. In addition, several other consequences are derived regarding graph connectivity and cycle statistics of the graph.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"337-360"},"PeriodicalIF":0.9,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142869195","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}

引用次数: 0

Short rainbow cycles for families of matchings and triangles

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-23 DOI: 10.1002/jgt.23183

He Guo

{"title":"Short rainbow cycles for families of matchings and triangles","authors":"He Guo","doi":"10.1002/jgt.23183","DOIUrl":"https://doi.org/10.1002/jgt.23183","url":null,"abstract":"A generalization of the famous Caccetta–Häggkvist conjecture, suggested by Aharoni, is that any family <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 \u0000 <mo>=</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}=({F}_{1},ldots ,{F}_{n})$</annotation>\u0000 </semantics></math> of sets of edges in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${K}_{n}$</annotation>\u0000 </semantics></math>, each of size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>, has a rainbow cycle of length at most <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⌈</mo>\u0000 \u0000 <mfrac>\u0000 <mi>n</mi>\u0000 \u0000 <mi>k</mi>\u0000 </mfrac>\u0000 \u0000 <mo>⌉</mo>\u0000 </mrow>\u0000 <annotation> $lceil frac{n}{k}rceil $</annotation>\u0000 </semantics></math>. In works by the author with Aharoni and by the author with Aharoni, Berger, Chudnovsky, and Zerbib, it was shown that asymptotically this can be improved to <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>log</mi>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $O(mathrm{log}n)$</annotation>\u0000 </se","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"325-336"},"PeriodicalIF":0.9,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23183","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868944","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}

引用次数: 0

On two problems of defective choosability of graphs

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-23 DOI: 10.1002/jgt.23182

Jie Ma, Rongxing Xu, Xuding Zhu

{"title":"On two problems of defective choosability of graphs","authors":"Jie Ma, Rongxing Xu, Xuding Zhu","doi":"10.1002/jgt.23182","DOIUrl":"https://doi.org/10.1002/jgt.23182","url":null,"abstract":"Given positive integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>≥</mo>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $pge k$</annotation>\u0000 </semantics></math>, and a nonnegative integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math>, we say a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>,</mo>\u0000 <mi>d</mi>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(k,d,p)$</annotation>\u0000 </semantics></math>-choosable if for every list assignment <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 <mi>L</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>∣</mo>\u0000 <mo>≥</mo>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $| L(v)| ge k$</annotation>\u0000 </semantics></math> for each <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>∈</mo>\u0000 <mi>V</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $vin V(G)$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 <msub>\u0000 <mo>⋃</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>∈</mo>\u0000 <mi>V</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </msub>\u0000 <mi>L</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"313-324"},"PeriodicalIF":0.9,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868925","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}

引用次数: 0

d $d$ -connectivity of the random graph with restricted budget

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-18 DOI: 10.1002/jgt.23180

Lyuben Lichev

{"title":"d\u0000 \u0000 $d$\u0000 -connectivity of the random graph with restricted budget","authors":"Lyuben Lichev","doi":"10.1002/jgt.23180","DOIUrl":"https://doi.org/10.1002/jgt.23180","url":null,"abstract":"In this short note, we consider a graph process recently introduced by Frieze, Krivelevich and Michaeli. In their model, the edges of the complete graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${K}_{n}$</annotation>\u0000 </semantics></math> are ordered uniformly at random and are then revealed consecutively to a player called Builder. At every round, Builder must decide if they accept the edge proposed at this round or not. We prove that, for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $dge 2$</annotation>\u0000 </semantics></math>, Builder can construct a spanning <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math>-connected graph after <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mi>log</mi>\u0000 \u0000 <mo> </mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $(1+o(1))nmathrm{log}unicode{x0200A}n/2$</annotation>\u0000 </semantics></math> rounds by accepting <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mi>d</mi>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"293-312"},"PeriodicalIF":0.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868961","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}

引用次数: 0

Finding dense minors using average degree 利用平均程度查找密集的未成年人

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-16 DOI: 10.1002/jgt.23169

Kevin Hendrey, Sergey Norin, Raphael Steiner, Jérémie Turcotte

{"title":"Finding dense minors using average degree","authors":"Kevin Hendrey, Sergey Norin, Raphael Steiner, Jérémie Turcotte","doi":"10.1002/jgt.23169","DOIUrl":"https://doi.org/10.1002/jgt.23169","url":null,"abstract":"Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible <math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow></math>-vertex minor in graphs of average degree at least <math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math>. We show that if <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> has average degree at least <math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math>, it contains a minor on <math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow></math> vertices with at least <math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msqrt>\u0000 <mn>2</mn>\u0000 </msqrt>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mfenced>\u0000 <mfrac>\u0000 <mi>t</mi>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mfenced>\u0000 </mrow></math> edges. We show that this cannot be improved beyond <math>\u0000 \u0000 <mrow>\u0000 <mfenced>\u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>4</mn>\u0000 </mfrac>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mfenced>\u0000 \u0000 <mfenced>\u0000 <mfrac>\u0000 <mi>t</mi>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mfenced>\u0000 </mrow></math>. Finally, for <math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"205-223"},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142707851","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}

引用次数: 0

Compatible powers of Hamilton cycles in dense graphs 密集图中汉密尔顿循环的相容幂

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-16 DOI: 10.1002/jgt.23178

Xiaohan Cheng, Jie Hu, Donglei Yang

{"title":"Compatible powers of Hamilton cycles in dense graphs","authors":"Xiaohan Cheng, Jie Hu, Donglei Yang","doi":"10.1002/jgt.23178","DOIUrl":"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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>=</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>V</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>E</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G=(V,E)$</annotation>\u0000 </semantics></math>, an incompatibility system <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}$</annotation>\u0000 </semantics></math> over <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a family <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <msub>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>v</mi>\u0000 </msub>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>V</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}={{{F}_{v}}}_{vin V}$</annotation>\u0000 </semantics></math> such that for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>V</mi>\u0000 </mrow>\u0000 <annotation> $vin V$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>v</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${F}_{v}$</annotation>\u0000 </semantics></math> is a family of edge pairs <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mrow>\u0000 <mi>e</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"257-273"},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","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}

引用次数: 0

Fractional factors and component factors in graphs with isolated toughness smaller than 1 孤立韧性小于 1 的图形中的分数因子和分量因子

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-16 DOI: 10.1002/jgt.23179

Isaak H. Wolf

{"title":"Fractional factors and component factors in graphs with isolated toughness smaller than 1","authors":"Isaak H. Wolf","doi":"10.1002/jgt.23179","DOIUrl":"10.1002/jgt.23179","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> be a simple graph and let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $n,m$</annotation>\u0000 </semantics></math> be two integers with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo><</mo>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo><</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $0lt mlt n$</annotation>\u0000 </semantics></math>. We prove that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>i</mi>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>S</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mfrac>\u0000 <mi>n</mi>\u0000 \u0000 <mi>m</mi>\u0000 </mfrac>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mi>S</mi>\u0000 \u0000 <mo>∣</mo>\u0000 </mrow>\u0000 <annotation> $iso(G-S)le frac{n}{m}| S| $</annotation>\u0000 </semantics></math> for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</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 </mrow>\u0000 <annotation> $Ssubset V(G)$</annotation>\u0000 </semantics></math> if and only if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"274-287"},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23179","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":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0