Journal of Graph Theory最新文献_第5页

Attainable bounds for algebraic connectivity and maximally connected regular graphs 代数连通性和最大连通正则图的可实现边界

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-24 DOI: 10.1002/jgt.23146

Geoffrey Exoo, Theodore Kolokolnikov, Jeanette Janssen, Timothy Salamon

{"title":"Attainable bounds for algebraic connectivity and maximally connected regular graphs","authors":"Geoffrey Exoo, Theodore Kolokolnikov, Jeanette Janssen, Timothy Salamon","doi":"10.1002/jgt.23146","DOIUrl":"10.1002/jgt.23146","url":null,"abstract":"We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon–Boppana–Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for well-known special cases. For the diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3-regular graphs, we find attainable graphs for all diameters <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> up to and including <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>9</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=9$</annotation>\u0000 </semantics></math> (the case of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>10</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=10$</annotation>\u0000 </semantics></math> is open). These graphs are extremely rare and also have high girth; for example, we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>7</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=7$</annotation>\u0000 </semantics></math>; all have girth 8. We also exhibit several infinite families attaining the upper bound with respect to diameter or girth. In particular, when <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math> is a power of prime, we construct a <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"522-549"},"PeriodicalIF":0.9,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23146","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504369","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 oriented m $m$ -semiregular representations of finite groups 关于有限群的定向 m $m$ 半圆代表

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-21 DOI: 10.1002/jgt.23145

Jia-Li Du, Yan-Quan Feng, Sejeong Bang

{"title":"On oriented \u0000 \u0000 \u0000 \u0000 m\u0000 \u0000 \u0000 $m$\u0000 -semiregular representations of finite groups","authors":"Jia-Li Du, Yan-Quan Feng, Sejeong Bang","doi":"10.1002/jgt.23145","DOIUrl":"10.1002/jgt.23145","url":null,"abstract":"A finite group <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> admits an oriented regular representation if there exists a Cayley digraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> such that it has no digons and its automorphism group is isomorphic to <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-semiregular representations using <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-Cayley digraphs. Given a finite group <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, an <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-Cayley digraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a digraph that has a group of automorphisms isomorphic to <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"485-508"},"PeriodicalIF":0.9,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504370","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

Highly connected triples and Mader's conjecture 高连接三元组和马德猜想

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-17 DOI: 10.1002/jgt.23144

Qinghai Liu, Kai Ying, Yanmei Hong

{"title":"Highly connected triples and Mader's conjecture","authors":"Qinghai Liu, Kai Ying, Yanmei Hong","doi":"10.1002/jgt.23144","DOIUrl":"https://doi.org/10.1002/jgt.23144","url":null,"abstract":"Mader proved that, for any tree <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>, every <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-connected graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>δ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $delta (G)ge 2{(k+m-1)}^{2}+m-1$</annotation>\u0000 </semantics></math> contains a subtree <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>T</mi>\u0000 \u0000 <mo>′</mo>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"478-484"},"PeriodicalIF":0.9,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142234910","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

A tight upper bound on the average order of dominating sets of a graph 图的支配集平均阶数的严格上限

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-17 DOI: 10.1002/jgt.23143

Iain Beaton, Ben Cameron

{"title":"A tight upper bound on the average order of dominating sets of a graph","authors":"Iain Beaton, Ben Cameron","doi":"10.1002/jgt.23143","DOIUrl":"https://doi.org/10.1002/jgt.23143","url":null,"abstract":"In this paper we study the average order of dominating sets in a graph, <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mstyle>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>avd</mtext>\u0000 <mspace></mspace>\u0000 </mstyle>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $,text{avd},(G)$</annotation>\u0000 </semantics></math>. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> without isolated vertices, <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>avd</mtext>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $,text{avd},(G)le 2n/3$</annotation>\u0000 </semantics></math>. Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>avd</mtext>\u0000 <mspace></mspace>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"463-477"},"PeriodicalIF":0.9,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23143","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142234963","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

Some results and problems on clique coverings of hypergraphs 关于超图的簇覆盖的一些结果和问题

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-11 DOI: 10.1002/jgt.23111

Vojtech Rödl, Marcelo Sales

{"title":"Some results and problems on clique coverings of hypergraphs","authors":"Vojtech Rödl, Marcelo Sales","doi":"10.1002/jgt.23111","DOIUrl":"https://doi.org/10.1002/jgt.23111","url":null,"abstract":"For a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-uniform hypergraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> we consider the parameter <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Θ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>F</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Theta }}(F)$</annotation>\u0000 </semantics></math>, the minimum size of a clique cover of the edge set of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math>. We derive bounds on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Θ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>F</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Theta }}(F)$</annotation>\u0000 </semantics></math> for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> belonging to various classes of hypergraphs.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"442-457"},"PeriodicalIF":0.9,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23111","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967877","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

Ramsey-type problems on induced covers and induced partitions toward the Gyárfás–Sumner conjecture 走向 Gyárfás-Sumner 猜想的诱导盖和诱导分区上的拉姆齐型问题

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-06 DOI: 10.1002/jgt.23124

Shuya Chiba, Michitaka Furuya

{"title":"Ramsey-type problems on induced covers and induced partitions toward the Gyárfás–Sumner conjecture","authors":"Shuya Chiba, Michitaka Furuya","doi":"10.1002/jgt.23124","DOIUrl":"https://doi.org/10.1002/jgt.23124","url":null,"abstract":"Gyárfás and Sumner independently conjectured that for every tree <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>, there exists a function <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msub>\u0000 \u0000 <mo>:</mo>\u0000 \u0000 <mi>N</mi>\u0000 \u0000 <mo>→</mo>\u0000 \u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation> ${f}_{T}:{mathbb{N}}to {mathbb{N}}$</annotation>\u0000 </semantics></math> such that every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>-free graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> satisfies <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>ω</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $chi (G)le {f}_{T}(omega (G))$</annotation>\u0000 </semantics></math>, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $chi (G)$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ω</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"419-441"},"PeriodicalIF":0.9,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967063","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

Kempe equivalent list colorings revisited Kempe 等价表着色再探讨

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-04 DOI: 10.1002/jgt.23142

Dibyayan Chakraborty, Carl Feghali, Reem Mahmoud

{"title":"Kempe equivalent list colorings revisited","authors":"Dibyayan Chakraborty, Carl Feghali, Reem Mahmoud","doi":"10.1002/jgt.23142","DOIUrl":"https://doi.org/10.1002/jgt.23142","url":null,"abstract":"A Kempe chain on colors <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> is a component of the subgraph induced by colors <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>. A Kempe change is the operation of interchanging the colors of some Kempe chains. For a list-assignment <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math> and an <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-coloring <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>φ</mi>\u0000 </mrow>\u0000 <annotation> $varphi $</annotation>\u0000 </semantics></math>, a Kempe change is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-valid for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>φ</mi>\u0000 </mrow>\u0000 <annotation> $varphi $</annotation>\u0000 </semantics></math> if performing the Kempe change yields another <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-coloring. Two <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-colorings are <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-equivalent if we can form one from the other by a sequence of <math>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"410-418"},"PeriodicalIF":0.9,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23142","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141966576","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 the maximum local mean order of sub- k $k$ -trees of a k $k$ -tree 关于 k $k$ 树的子 k $k$ 树的最大局部平均阶数

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-02 DOI: 10.1002/jgt.23128

Zhuo Li, Tianlong Ma, Fengming Dong, Xian'an Jin

{"title":"On the maximum local mean order of sub-\u0000 \u0000 \u0000 k\u0000 \u0000 $k$\u0000 -trees of a \u0000 \u0000 \u0000 k\u0000 \u0000 $k$\u0000 -tree","authors":"Zhuo Li, Tianlong Ma, Fengming Dong, Xian'an Jin","doi":"10.1002/jgt.23128","DOIUrl":"10.1002/jgt.23128","url":null,"abstract":"For a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-tree <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>, a generalization of a tree, the local mean order of sub-<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> is the average order of sub-<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> containing a given <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-clique. The problem whether the maximum local mean order of a tree (i.e., a 1-tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-tree <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>, does the maximum local mean order of sub-<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees containing a given <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-clique occur at a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-clique that is not a major </span","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"393-409"},"PeriodicalIF":0.9,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141273372","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

Counting connected partitions of graphs 计算图的连接分区

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-02 DOI: 10.1002/jgt.23127

Yair Caro, Balázs Patkós, Zsolt Tuza, Máté Vizer

{"title":"Counting connected partitions of graphs","authors":"Yair Caro, Balázs Patkós, Zsolt Tuza, Máté Vizer","doi":"10.1002/jgt.23127","DOIUrl":"https://doi.org/10.1002/jgt.23127","url":null,"abstract":"Motivated by the theorem of Győri and Lovász, we consider the following problem. For a connected graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> vertices and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> edges determine the number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>,</mo>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $P(G,k)$</annotation>\u0000 </semantics></math> of unordered solutions of positive integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mo>∑</mo>\u0000 <mrow>\u0000 <mi>i</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mi>k</mi>\u0000 </msubsup>\u0000 <msub>\u0000 <mi>m</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> ${sum }_{i=1}^{k}{m}_{i}=m$</annotation>\u0000 </semantics></math> such that every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>m</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${m}_{i}$</annotation>\u0000 </semantics></math> is realized by a connected subgraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${H}_{i}$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>m</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${m}_{i}$</annotation>\u0000 </s","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"381-392"},"PeriodicalIF":0.9,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967053","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

The maximum number of maximum generalized 4-independent sets in trees 树中最大广义 4 个独立集合的最大数量

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-30 DOI: 10.1002/jgt.23122

Pingshan Li, Min Xu

{"title":"The maximum number of maximum generalized 4-independent sets in trees","authors":"Pingshan Li, Min Xu","doi":"10.1002/jgt.23122","DOIUrl":"10.1002/jgt.23122","url":null,"abstract":"A generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-independent set is a set of vertices such that the induced subgraph contains no trees with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-vertices, and the generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-independence number is the cardinality of a maximum <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-independent set in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. Zito proved that the maximum number of maximum generalized 2-independent sets in a tree of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mfrac>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${2}^{frac{n-3}{2}}$</annotation>\u0000 </semantics></math> if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> is odd, and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mfrac>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> ${2}^{frac{n-2}{2}}+1$</annotation>\u0000 </semantics></math> if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"359-380"},"PeriodicalIF":0.9,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195007","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