Journal of Graph Theory最新文献

{"title":"Inclusion chromatic index of random graphs","authors":"Jakub Kwaśny, Jakub Przybyło","doi":"10.1002/jgt.23088","DOIUrl":"https://doi.org/10.1002/jgt.23088","url":null,"abstract":"Erdős and Wilson proved in 1977 that almost all graphs have chromatic index equal to their maximum degree. In 2001 Balister extended this result and proved that the same number of colours is almost always sufficient if we additionally demand the distinctness of the sets of colours incident with any two vertices. We study a stronger condition and show that one more colour is almost always sufficient and necessary if the inclusion of these sets is forbidden for any pair of adjacent vertices. We also settle the value of a more restrictive graph invariant for almost all graphs, where inclusion is forbidden for all pairs of vertices, which necessitates one more colour for graphs of even order.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139838073","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":"Degree criteria and stability for independent transversals","authors":"Penny Haxell,&nbsp;Ronen Wdowinski","doi":"10.1002/jgt.23085","DOIUrl":"https://doi.org/10.1002/jgt.23085","url":null,"abstract":"<p>An <i>independent transversal</i> (IT) in a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with a given vertex partition <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math> is an independent set of vertices of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> (i.e., it induces no edges), that consists of one vertex from each part (<i>block</i>) of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math>. Over the years, various criteria have been established that guarantee the existence of an IT, often given in terms of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math> being <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-<i>thick</i>, meaning all blocks have size at least <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>. One such result, obtained recently by Wanless and Wood, is based on the <i>maximum average block degree</i> <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>,</mo>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mi>max</mi>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mo>∑</mo>\u0000 <mrow>\u0000 <mi>u</mi>\u0000 <mo>∈</mo>\u0000 <mi>U</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mi>d</mi>\u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23085","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140552914","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":"Inclusion chromatic index of random graphs","authors":"Jakub Kwaśny,&nbsp;Jakub Przybyło","doi":"10.1002/jgt.23088","DOIUrl":"10.1002/jgt.23088","url":null,"abstract":"<p>Erdős and Wilson proved in 1977 that almost all graphs have chromatic index equal to their maximum degree. In 2001 Balister extended this result and proved that the same number of colours is almost always sufficient if we additionally demand the distinctness of the sets of colours incident with any two vertices. We study a stronger condition and show that one more colour is almost always sufficient and necessary if the inclusion of these sets is forbidden for any pair of adjacent vertices. We also settle the value of a more restrictive graph invariant for almost all graphs, where inclusion is forbidden for all pairs of vertices, which necessitates one more colour for graphs of even order.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139778461","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":"On tree decompositions whose trees are minors","authors":"Pablo Blanco,&nbsp;Linda Cook,&nbsp;Meike Hatzel,&nbsp;Claire Hilaire,&nbsp;Freddie Illingworth,&nbsp;Rose McCarty","doi":"10.1002/jgt.23083","DOIUrl":"10.1002/jgt.23083","url":null,"abstract":"<p>In 2019, Dvořák asked whether every connected graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> has a tree decomposition <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>B</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(T,{rm{ {mathcal B} }})$</annotation>\u0000 </semantics></math> so that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> is a subgraph of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> and the width of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>B</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(T,{rm{ {mathcal B} }})$</annotation>\u0000 </semantics></math> is bounded by a function of the treewidth of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. We prove that this is false, even when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> has treewidth 2 and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> is allowed to be a minor of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23083","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139762312","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":"Minimum degree stability of \u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 2\u0000 k\u0000 +\u0000 1\u0000 \u0000 \u0000 \u0000 ${C}_{2k+1}$\u0000 -free graphs","authors":"Xiaoli Yuan,&nbsp;Yuejian Peng","doi":"10.1002/jgt.23086","DOIUrl":"10.1002/jgt.23086","url":null,"abstract":"<p>We consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${C}_{2k+1}$</annotation>\u0000 </semantics></math>-free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>5</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mtext>…</mtext>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation> ${{C}_{3},{C}_{5},ldots ,{C}_{2k+1}}$</annotation>\u0000 </semantics></math>-free graph on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> vertices has minimum degree greater than <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mfrac>\u0000 <mn>2</mn>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mfrac>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $frac{2}{2k+3}n$</annotation>\u0000 </semantics></math>, then it is bipartite. Häggkvist showed that for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139762501","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":"Integer flows on triangularly connected signed graphs","authors":"Liangchen Li,&nbsp;Chong Li,&nbsp;Rong Luo,&nbsp;Cun-Quan Zhang","doi":"10.1002/jgt.23076","DOIUrl":"10.1002/jgt.23076","url":null,"abstract":"<p>A triangle-path in a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a sequence of distinct triangles <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mi>m</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${T}_{1},{T}_{2},ldots ,{T}_{m}$</annotation>\u0000 </semantics></math> in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> such that for any <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>i</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>j</mi>\u0000 </mrow>\u0000 <annotation> $i,j$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>i</mi>\u0000 \u0000 <mo>&lt;</mo>\u0000 \u0000 <mi>j</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $1le ilt jle m$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>E</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∩</mo>\u0000 \u0000 <mi>E</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <msub>\u0000 <mi>T</mi>\u0000 <mrow>\u0000 <mi>i</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</m","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139762709","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 note on the width of sparse random graphs","authors":"Tuan Anh Do,&nbsp;Joshua Erde,&nbsp;Mihyun Kang","doi":"10.1002/jgt.23081","DOIUrl":"https://doi.org/10.1002/jgt.23081","url":null,"abstract":"<p>In this note, we consider the width of a supercritical random graph according to some commonly studied width measures. We give short, direct proofs of results of Lee, Lee and Oum, and of Perarnau and Serra, on the rank- and tree-width of the random graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G(n,p)$</annotation>\u0000 </semantics></math> when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mfrac>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>+</mo>\u0000 <mi>ϵ</mi>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation> $p=frac{1+epsilon }{n}$</annotation>\u0000 </semantics></math> for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ϵ</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation> $epsilon gt 0$</annotation>\u0000 </semantics></math> constant. Our proofs avoid the use of black box results on the expansion properties of the giant component in this regime, and so as a further benefit we obtain explicit bounds on the dependence of these results on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ϵ</mi>\u0000 </mrow>\u0000 <annotation> $epsilon $</annotation>\u0000 </semantics></math>. Finally, we also consider the width of the random graph in the <i>weakly supercritical regime</i>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ϵ</mi>\u0000 <mo>=</mo>\u0000 <mi>o</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $epsilon =o(1)$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>ϵ</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <mi>n</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation> ${epsilon }^{3}nto infty $</annotati","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23081","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140552871","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":"A lower bound for the complex flow number of a graph: A geometric approach","authors":"Davide Mattiolo,&nbsp;Giuseppe Mazzuoccolo,&nbsp;Jozef Rajník,&nbsp;Gloria Tabarelli","doi":"10.1002/jgt.23075","DOIUrl":"https://doi.org/10.1002/jgt.23075","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 <mo>≥</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $rge 2$</annotation>\u0000 </semantics></math> be a real number. A complex nowhere-zero <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>-flow on a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is an orientation of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> together with an assignment <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>φ</mi>\u0000 <mo>:</mo>\u0000 <mi>E</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>→</mo>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> $varphi :E(G)to {mathbb{C}}$</annotation>\u0000 </semantics></math> such that, for all <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>e</mi>\u0000 <mo>∈</mo>\u0000 <mi>E</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $ein E(G)$</annotation>\u0000 </semantics></math>, the Euclidean norm of the complex number <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>φ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>e</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $varphi (e)$</annotation>\u0000 </semantics></math> lies in the interval <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>r</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation> $[1,r-1]$</annotation>\u0000 </semantics></math> and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph <span></span><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":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140552856","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":"On two cycles of consecutive even lengths","authors":"Jun Gao,&nbsp;Binlong Li,&nbsp;Jie Ma,&nbsp;Tianying Xie","doi":"10.1002/jgt.23074","DOIUrl":"10.1002/jgt.23074","url":null,"abstract":"<p>Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two. We prove the following average degree counterpart that every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with at least <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mfrac>\u0000 <mn>5</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $frac{5}{2}(n-1)$</annotation>\u0000 </semantics></math> edges, unless <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 \u0000 <mo>|</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $4|(n-1)$</annotation>\u0000 </semantics></math> and every block of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a clique <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>5</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${K}_{5}$</annotation>\u0000 </semantics></math>, contains two cycles of consecutive even lengths. Our proof is mainly based on structural analysis, and a crucial step which may be of independent interest shows that the same conclusion holds for every 3-connected graph with at least six vertices. This solves a special case of a conjecture of Verstraëte. The quantitative bound is tight and also provides the optimal extremal number for cycles of length two modulo four.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139560900","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 proof of Frankl–Kupavskii's conjecture on edge-union condition","authors":"Hongliang Lu,&nbsp;Xuechun Zhang","doi":"10.1002/jgt.23073","DOIUrl":"10.1002/jgt.23073","url":null,"abstract":"<p>A 3-graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}$</annotation>\u0000 </semantics></math> is <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>U</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>s</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $U(s,2s+1)$</annotation>\u0000 </semantics></math> if for any <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $s$</annotation>\u0000 </semantics></math> edges <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>e</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>e</mi>\u0000 \u0000 <mi>s</mi>\u0000 </msub>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>E</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>F</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${e}_{1},ldots ,{e}_{s}in E({rm{ {mathcal F} }})$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>∪</mo>\u0000 \u0000 <mtext>⋯</mtext>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139412688","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}