Journal of Graph Theory最新文献_第10页

Concentration of hitting times in Erdős-Rényi graphs 厄尔多斯-雷尼图中命中时间的浓度

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-12 DOI: 10.1002/jgt.23119

Andrea Ottolini, Stefan Steinerberger

{"title":"Concentration of hitting times in Erdős-Rényi graphs","authors":"Andrea Ottolini, Stefan Steinerberger","doi":"10.1002/jgt.23119","DOIUrl":"10.1002/jgt.23119","url":null,"abstract":"We consider Erdős-Rényi graphs <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> for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 <mo><</mo>\u0000 <mi>p</mi>\u0000 <mo><</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $0lt plt 1$</annotation>\u0000 </semantics></math> fixed and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation> $nto infty $</annotation>\u0000 </semantics></math> and study the expected number of steps, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mrow>\u0000 <mi>w</mi>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${H}_{wv}$</annotation>\u0000 </semantics></math>, that a random walk started in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>w</mi>\u0000 </mrow>\u0000 <annotation> $w$</annotation>\u0000 </semantics></math> needs to first arrive in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math>. A natural guess is that an Erdős-Rényi random graph is so homogeneous that it does not really distinguish between vertices and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mrow>\u0000 <mi>w</mi>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>+</mo>\u0000 <mi>o</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> ${H}_{wv}=(1+o(1))n$</annotation>\u0000 </sema","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"245-262"},"PeriodicalIF":0.9,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933192","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

Ubiquity of oriented rays 定向射线的普遍性

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-12 DOI: 10.1002/jgt.23114

Florian Gut, Thilo Krill, Florian Reich

{"title":"Ubiquity of oriented rays","authors":"Florian Gut, Thilo Krill, Florian Reich","doi":"10.1002/jgt.23114","DOIUrl":"10.1002/jgt.23114","url":null,"abstract":"A digraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> is called ubiquitous if every digraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> that contains <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> vertex-disjoint copies of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>∈</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation> $kin {mathbb{N}}$</annotation>\u0000 </semantics></math> also contains infinitely many vertex-disjoint copies of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math>. We characterise which digraphs with rays as underlying undirected graphs are ubiquitous.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"200-211"},"PeriodicalIF":0.9,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23114","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933310","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

Density of 3-critical signed graphs 三临界有符号图形的密度

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-12 DOI: 10.1002/jgt.23117

Laurent Beaudou, Penny Haxell, Kathryn Nurse, Sagnik Sen, Zhouningxin Wang

{"title":"Density of 3-critical signed graphs","authors":"Laurent Beaudou, Penny Haxell, Kathryn Nurse, Sagnik Sen, Zhouningxin Wang","doi":"10.1002/jgt.23117","DOIUrl":"10.1002/jgt.23117","url":null,"abstract":"We say that a signed graph is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-critical if it is not <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-colorable but every one of its proper subgraphs is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every 3-critical signed graph on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> vertices has at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mfrac>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation> $frac{3n-1}{2}$</annotation>\u0000 </semantics></math> edges, and that this bound is asymptotically tight. It follows that every signed planar or projective-planar graph of girth at least 6 is (circular) 3-colorable, and for the projective-planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>C</mi>\u0000 <mn>3</mn>\u0000 <mo>*</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation> ${C}_{3}^{* }$</annotation>\u0000 </semantics></math>, which is the positive triangle augmented with a negative loop on each vertex.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"212-239"},"PeriodicalIF":0.9,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933031","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

Cutting a tree with subgraph complementation is hard, except for some small trees 用子图互补法切割一棵树是很难的，除非是一些小树

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-09 DOI: 10.1002/jgt.23112

Dhanyamol Antony, Sagartanu Pal, R. B. Sandeep, R. Subashini

{"title":"Cutting a tree with subgraph complementation is hard, except for some small trees","authors":"Dhanyamol Antony, Sagartanu Pal, R. B. Sandeep, R. Subashini","doi":"10.1002/jgt.23112","DOIUrl":"10.1002/jgt.23112","url":null,"abstract":"For a graph property <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Π</mi>\u0000 </mrow>\u0000 <annotation> ${rm{Pi }}$</annotation>\u0000 </semantics></math>, Subgraph Complementation to <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Π</mi>\u0000 </mrow>\u0000 <annotation> ${rm{Pi }}$</annotation>\u0000 </semantics></math> is the problem to find whether there is a subset <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> $S$</annotation>\u0000 </semantics></math> of vertices of the input graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> such that modifying <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> by complementing the subgraph induced by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> $S$</annotation>\u0000 </semantics></math> results in a graph satisfying the property <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Π</mi>\u0000 </mrow>\u0000 <annotation> ${rm{Pi }}$</annotation>\u0000 </semantics></math>. We prove that the problem of Subgraph Complementation to <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>-free graphs is NP-Complete, for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> being a tree, except for 41 trees of at most 13 vertices (a graph is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>-free if it does not contain any induced copies of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>). This result, along with the four known polynomial-time solvable cases (when <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> is a path on at most four vertices), leaves behind 37 open cases. Further, we prove that these hard problems do ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"126-168"},"PeriodicalIF":0.9,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933033","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

Turán number of the odd-ballooning of complete bipartite graphs 完整双方位图奇数包络的图兰数

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-09 DOI: 10.1002/jgt.23118

Xing Peng, Mengjie Xia

{"title":"Turán number of the odd-ballooning of complete bipartite graphs","authors":"Xing Peng, Mengjie Xia","doi":"10.1002/jgt.23118","DOIUrl":"10.1002/jgt.23118","url":null,"abstract":"Given a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>, the Turán number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>ex</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{ex}(n,L)$</annotation>\u0000 </semantics></math> is the maximum possible number of edges in an <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-free graph. The study of Turán number of graphs is a central topic in extremal graph theory. Although the celebrated Erdős-Stone-Simonovits theorem gives the asymptotic value of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>ex</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{ex}(n,L)$</annotation>\u0000 </semantics></math> for nonbipartite <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>, it is challenging in general to determine the exact value of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>ex</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{ex}(n,L)$</annotation>\u0000 </semantics></math> for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>L</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation> $chi (L)ge 3$</annotation>\u0000 </semant","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"181-199"},"PeriodicalIF":0.9,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933026","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

On induced subgraph of Cartesian product of paths 论路径笛卡尔积的诱导子图

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-09 DOI: 10.1002/jgt.23116

Jiasheng Zeng, Xinmin Hou

{"title":"On induced subgraph of Cartesian product of paths","authors":"Jiasheng Zeng, Xinmin Hou","doi":"10.1002/jgt.23116","DOIUrl":"10.1002/jgt.23116","url":null,"abstract":"Chung et al. constructed an induced subgraph of the hypercube <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>Q</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${Q}^{n}$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <msup>\u0000 <mi>Q</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msup>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $alpha ({Q}^{n})+1$</annotation>\u0000 </semantics></math> vertices and with maximum degree smaller than <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⌈</mo>\u0000 \u0000 <msqrt>\u0000 <mi>n</mi>\u0000 </msqrt>\u0000 \u0000 <mo>⌉</mo>\u0000 </mrow>\u0000 <annotation> $lceil sqrt{n}rceil $</annotation>\u0000 </semantics></math>. Subsequently, Huang proved the Sensitivity Conjecture by demonstrating that the maximum degree of such an induced subgraph of hypercube <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>Q</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${Q}^{n}$</annotation>\u0000 </semantics></math> is at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⌈</mo>\u0000 \u0000 <msqrt>\u0000 <mi>n</mi>\u0000 </msqrt>\u0000 \u0000 <mo>⌉</mo>\u0000 </mrow>\u0000 <annotation> $lceil sqrt{n}rceil $</annotation>\u0000 </semantics></math>, and posed the question: Given a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $f(G)$</annotation>\u0000 </semantics></math> be the minimum of the maximum degree of an induced subgraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"169-180"},"PeriodicalIF":0.9,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941656","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

Sharp threshold for embedding balanced spanning trees in random geometric graphs 在随机几何图中嵌入平衡生成树的锐阈值

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-09 DOI: 10.1002/jgt.23106

Alberto Espuny Díaz, Lyuben Lichev, Dieter Mitsche, Alexandra Wesolek

{"title":"Sharp threshold for embedding balanced spanning trees in random geometric graphs","authors":"Alberto Espuny Díaz, Lyuben Lichev, Dieter Mitsche, Alexandra Wesolek","doi":"10.1002/jgt.23106","DOIUrl":"10.1002/jgt.23106","url":null,"abstract":"A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph <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>r</mi>\u0000 <mo>,</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${mathscr{G}}(n,r,d)$</annotation>\u0000 </semantics></math>. In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"107-125"},"PeriodicalIF":0.9,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933029","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

Local degree conditions for K 9 ${K}_{9}$ -minors in graphs 图中 K9 ${K}_{9}$ 未成数的局部度条件

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-08 DOI: 10.1002/jgt.23110

Takashige Akiyama

{"title":"Local degree conditions for \u0000 \u0000 \u0000 \u0000 K\u0000 9\u0000 \u0000 \u0000 ${K}_{9}$\u0000 -minors in graphs","authors":"Takashige Akiyama","doi":"10.1002/jgt.23110","DOIUrl":"10.1002/jgt.23110","url":null,"abstract":"We prove that if each edge of a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> belongs to at least seven triangles, then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> contains a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>9</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${K}_{9}$</annotation>\u0000 </semantics></math>-minor or contains <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${K}_{1,2,2,2,2,2}$</annotation>\u0000 </semantics></math> as an induced subgraph. This result was conjectured by Albar and Gonçalves in 2018. Moreover, we apply this result to study the stress freeness of graphs.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"70-94"},"PeriodicalIF":0.9,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933027","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

Spanning even trees of graphs 图的跨偶数树

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-08 DOI: 10.1002/jgt.23115

Bill Jackson, Kiyoshi Yoshimoto

{"title":"Spanning even trees of graphs","authors":"Bill Jackson, Kiyoshi Yoshimoto","doi":"10.1002/jgt.23115","DOIUrl":"10.1002/jgt.23115","url":null,"abstract":"A tree <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> is said to be even if all pairs of vertices of degree one in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> are joined by a path of even length. We conjecture that every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>-regular nonbipartite connected graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> has a spanning even tree and verify this conjecture when <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> has a 2-factor. Well-known results of Petersen and Hanson et al. imply that the only remaining unsolved case is when <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math> is odd and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> has at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math> bridges. We investigate this case further and propose some related conjectures.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"95-106"},"PeriodicalIF":0.9,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933030","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

Graphs with at most two moplexes 最多有两个横线的图形

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-05-08 DOI: 10.1002/jgt.23102

Clément Dallard, Robert Ganian, Meike Hatzel, Matjaž Krnc, Martin Milanič

{"title":"Graphs with at most two moplexes","authors":"Clément Dallard, Robert Ganian, Meike Hatzel, Matjaž Krnc, Martin Milanič","doi":"10.1002/jgt.23102","DOIUrl":"10.1002/jgt.23102","url":null,"abstract":"A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes, such as interval graphs, claw-free, and diamond-free graphs. However, while every noncomplete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, in part, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are intervals. In this work, we initiate an investigation of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-moplex graphs, which are defined as graphs containing at most <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> moplexes. Of particular interest is the smallest nontrivial case <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $k=2$</annotation>\u0000 </semantics></math>, which forms a counterpart to the class of interval graphs. As our main structural result, we show that, when restricted to connected graphs, the class of 2-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity-theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely, \u0000Graph Isomorphism and \u0000Max-Cut. On the other hand, we prove that every connected 2-moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"38-69"},"PeriodicalIF":0.9,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23102","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933032","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