Journal of Graph Theory最新文献

{"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":"<p>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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"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":"<p>A finite group <span></span><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 <i>oriented regular representation</i> if there exists a Cayley digraph of <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. Let <span></span><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 <span></span><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 <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-<i>Cayley digraph</i> of <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"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":"<p>Mader proved that, for any tree <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>, every <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with <span></span><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 <span></span><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":null,"pages":null},"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}

{"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":"<p>In this paper we study the average order of dominating sets in a graph, <span></span><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 <span></span><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 <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> without isolated vertices, <span></span><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 <span></span><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":null,"pages":null},"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}

{"title":"Some results and problems on clique coverings of hypergraphs","authors":"Vojtech Rödl,&nbsp;Marcelo Sales","doi":"10.1002/jgt.23111","DOIUrl":"https://doi.org/10.1002/jgt.23111","url":null,"abstract":"<p>For a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-uniform hypergraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> we consider the parameter <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math>. We derive bounds on <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> belonging to various classes of hypergraphs.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"title":"Ramsey-type problems on induced covers and induced partitions toward the Gyárfás–Sumner conjecture","authors":"Shuya Chiba,&nbsp;Michitaka Furuya","doi":"10.1002/jgt.23124","DOIUrl":"https://doi.org/10.1002/jgt.23124","url":null,"abstract":"<p>Gyárfás and Sumner independently conjectured that for every tree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>, there exists a function <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>-free graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> satisfies <span></span><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 <span></span><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 <span></span><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":null,"pages":null},"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}

{"title":"Kempe equivalent list colorings revisited","authors":"Dibyayan Chakraborty,&nbsp;Carl Feghali,&nbsp;Reem Mahmoud","doi":"10.1002/jgt.23142","DOIUrl":"https://doi.org/10.1002/jgt.23142","url":null,"abstract":"<p>A <i>Kempe chain</i> on colors <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <annotation> $a$</annotation>\u0000 </semantics></math> and <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <annotation> $a$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $b$</annotation>\u0000 </semantics></math>. A <i>Kempe change</i> is the operation of interchanging the colors of some Kempe chains. For a list-assignment <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math> and an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-coloring <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>φ</mi>\u0000 </mrow>\u0000 <annotation> $varphi $</annotation>\u0000 </semantics></math>, a Kempe change is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-<i>valid</i> for <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-coloring. Two <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-colorings are <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-<i>equivalent</i> if we can form one from the other by a sequence of <span></span><math>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"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,&nbsp;Tianlong Ma,&nbsp;Fengming Dong,&nbsp;Xian'an Jin","doi":"10.1002/jgt.23128","DOIUrl":"10.1002/jgt.23128","url":null,"abstract":"<p>For a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-tree <span></span><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-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> is the average order of sub-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> containing a given <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-tree <span></span><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-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees containing a given <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-clique occur at a <span></span><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></span","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"title":"Counting connected partitions of graphs","authors":"Yair Caro,&nbsp;Balázs Patkós,&nbsp;Zsolt Tuza,&nbsp;Máté Vizer","doi":"10.1002/jgt.23127","DOIUrl":"https://doi.org/10.1002/jgt.23127","url":null,"abstract":"<p>Motivated by the theorem of Győri and Lovász, we consider the following problem. For a connected graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> vertices and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> edges determine the number <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with <span></span><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":null,"pages":null},"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}

{"title":"The maximum number of maximum generalized 4-independent sets in trees","authors":"Pingshan Li,&nbsp;Min Xu","doi":"10.1002/jgt.23122","DOIUrl":"10.1002/jgt.23122","url":null,"abstract":"<p>A generalized <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-vertices, and the generalized <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-independent set in <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> is <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> is odd, and <span></span><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 <span></span><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":null,"pages":null},"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}