Journal of Graph Theory最新文献

{"title":"Planar graphs having no cycle of length 4, 7, or 9 are DP-3-colorable","authors":"Yingli Kang,&nbsp;Ligang Jin,&nbsp;Xuding Zhu","doi":"10.1002/jgt.23123","DOIUrl":"10.1002/jgt.23123","url":null,"abstract":"<p>This paper proves that every planar graph having no cycle of length 4, 7, or 9 is DP-3-colorable.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141111900","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":"Minimum-degree conditions for rainbow triangles","authors":"Victor Falgas-Ravry,&nbsp;Klas Markström,&nbsp;Eero Räty","doi":"10.1002/jgt.23109","DOIUrl":"10.1002/jgt.23109","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>≔</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${bf{G}}:= ({G}_{1},{G}_{2},{G}_{3})$</annotation>\u0000 </semantics></math> be a triple of graphs on a common vertex set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>V</mi>\u0000 </mrow>\u0000 <annotation> $V$</annotation>\u0000 </semantics></math> of size <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>. A rainbow triangle in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> ${bf{G}}$</annotation>\u0000 </semantics></math> is a triple of edges <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $({e}_{1},{e}_{2},{e}_{3","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23109","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061415","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":"Bounding the number of odd paths in planar graphs via convex optimization","authors":"Asaf Cohen Antonir,&nbsp;Asaf Shapira","doi":"10.1002/jgt.23120","DOIUrl":"10.1002/jgt.23120","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>N</mi>\u0000 \u0000 <mi>P</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>H</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${N}_{{mathscr{P}}}(n,H)$</annotation>\u0000 </semantics></math> denote the maximum number of copies of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> in an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> vertex planar graph. The problem of bounding this function for various graphs <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> has been extensively studied since the 70's. A special case that received a lot of attention recently is when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> is the path on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $2m+1$</annotation>\u0000 </semantics></math> vertices, denoted <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>P</mi>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${P}_{2m+1}$</annotation>\u0000 </semantics></math>. Our main result in this paper is that\u0000\u0000 </p><p>This improves upon the previously best known ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23120","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061351","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 more accurate view of the Flat Wall Theorem","authors":"Ignasi Sau,&nbsp;Giannos Stamoulis,&nbsp;Dimitrios M. Thilikos","doi":"10.1002/jgt.23121","DOIUrl":"10.1002/jgt.23121","url":null,"abstract":"<p>We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in future algorithmic applications.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23121","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061358","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":"Ubiquity of oriented rays","authors":"Florian Gut,&nbsp;Thilo Krill,&nbsp;Florian Reich","doi":"10.1002/jgt.23114","DOIUrl":"10.1002/jgt.23114","url":null,"abstract":"<p>A digraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> is called <i>ubiquitous</i> if every digraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> that contains <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> vertex-disjoint copies of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> for every <span></span><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 <span></span><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.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"title":"Concentration of hitting times in Erdős-Rényi graphs","authors":"Andrea Ottolini,&nbsp;Stefan Steinerberger","doi":"10.1002/jgt.23119","DOIUrl":"10.1002/jgt.23119","url":null,"abstract":"<p>We consider Erdős-Rényi graphs <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> for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 <mo>&lt;</mo>\u0000 <mi>p</mi>\u0000 <mo>&lt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $0lt plt 1$</annotation>\u0000 </semantics></math> fixed and <span></span><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, <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>w</mi>\u0000 </mrow>\u0000 <annotation> $w$</annotation>\u0000 </semantics></math> needs to first arrive in <span></span><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 <span></span><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":null,"pages":null},"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}

{"title":"Density of 3-critical signed graphs","authors":"Laurent Beaudou,&nbsp;Penny Haxell,&nbsp;Kathryn Nurse,&nbsp;Sagnik Sen,&nbsp;Zhouningxin Wang","doi":"10.1002/jgt.23117","DOIUrl":"10.1002/jgt.23117","url":null,"abstract":"<p>We say that a signed graph is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-<i>critical</i> if it is not <span></span><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 <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> vertices has at least <span></span><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 <span></span><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.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"title":"Cutting a tree with subgraph complementation is hard, except for some small trees","authors":"Dhanyamol Antony,&nbsp;Sagartanu Pal,&nbsp;R. B. Sandeep,&nbsp;R. Subashini","doi":"10.1002/jgt.23112","DOIUrl":"10.1002/jgt.23112","url":null,"abstract":"<p>For a graph property <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Π</mi>\u0000 </mrow>\u0000 <annotation> ${rm{Pi }}$</annotation>\u0000 </semantics></math>, Subgraph Complementation to <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> $S$</annotation>\u0000 </semantics></math> of vertices of the input graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> such that modifying <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> by complementing the subgraph induced by <span></span><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 <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>-free graphs is NP-Complete, for <span></span><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 <span></span><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 <span></span><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 <span></span><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":null,"pages":null},"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}

{"title":"Turán number of the odd-ballooning of complete bipartite graphs","authors":"Xing Peng,&nbsp;Mengjie Xia","doi":"10.1002/jgt.23118","DOIUrl":"10.1002/jgt.23118","url":null,"abstract":"<p>Given a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>, the Turán number <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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":null,"pages":null},"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}

{"title":"On induced subgraph of Cartesian product of paths","authors":"Jiasheng Zeng,&nbsp;Xinmin Hou","doi":"10.1002/jgt.23116","DOIUrl":"10.1002/jgt.23116","url":null,"abstract":"<p>Chung et al. constructed an induced subgraph of the hypercube <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, let <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-09","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}