Journal of Graph Theory最新文献

{"title":"Acyclic graphs with at least 2ℓ + 1 vertices are ℓ‐recognizable","authors":"A. Kostochka, M. Nahvi, D. West, Dara Zirlin","doi":"10.1002/jgt.23027","DOIUrl":"https://doi.org/10.1002/jgt.23027","url":null,"abstract":"The ‐deck of an ‐vertex graph is the multiset of subgraphs obtained from it by deleting vertices. A family of ‐vertex graphs is ‐recognizable if every graph having the same ‐deck as a graph in the family is also in the family. We prove that the family of ‐vertex graphs with no cycles is ‐recognizable when (except for ). As a consequence, the family of ‐vertex trees is ‐recognizable when and . It is known that this fails when .","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42419426","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":"Rooted minors and locally spanning subgraphs","authors":"Thomas Böhme, Jochen Harant, Matthias Kriesell, Samuel Mohr, Jens M. Schmidt","doi":"10.1002/jgt.23012","DOIUrl":"10.1002/jgt.23012","url":null,"abstract":"<p>Results on the existence of various types of spanning subgraphs of graphs are milestones in structural graph theory and have been diversified in several directions. In the present paper, we consider “local” versions of such statements. In 1966, for instance, D. W. Barnette proved that a 3-connected planar graph contains a spanning tree of maximum degree at most 3. A local translation of this statement is that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a planar graph, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math> is a subset of specified vertices of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math> cannot be separated in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> by removing two or fewer vertices of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> has a tree of maximum degree at most 3 containing all vertices of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math>. Our results constitute a general machinery for strengthening statements about <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-connected graphs (for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $1le kle 4$</annotation>\u0000 </semantics></math>) to locally spanning versions, that is, subgraphs containing a set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>⊆</mo>\u0000 \u0000 <mi>V</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"105 2","pages":"209-229"},"PeriodicalIF":0.9,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43546595","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":"Minimal bricks with the maximum number of edges","authors":"Xing Feng, Weigen Yan","doi":"10.1002/jgt.23026","DOIUrl":"https://doi.org/10.1002/jgt.23026","url":null,"abstract":"A 3‐connected graph is a brick if, after the removal of any two distinct vertices, the resulting graph has a perfect matching. A brick is minimal if, for every edge , deleting results in a graph that is not a brick. Norine and Thomas proved that every minimal brick with vertices, which is distinct from the prism or the wheel on four, six, or eight vertices, has at most edges. In this paper, we characterize the extremal minimal bricks with vertices that meet this upper bound, and we prove that the number of extremal graphs equals if , 5 if , 10 if and 0 if , respectively.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44211939","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 average order of a connected induced subgraph of a graph and union‐intersection systems","authors":"A. Vince","doi":"10.1002/jgt.23024","DOIUrl":"https://doi.org/10.1002/jgt.23024","url":null,"abstract":"Because connectivity is such a basic concept in graph theory, extremal problems concerning the average order of the connected induced subgraphs of a graph have been of notable interest. A particularly resistant open problem is whether or not, for a connected graph of order , all of whose vertices have degree at least 3, this average is at least . It is shown in this paper that if is a connected, vertex transitive graph, then the average order of the connected induced subgraphs of is at least .The extremal graph theory problems mentioned above lead to a broader theory. The concept of a Union‐Intersection System (UIS)  is introduced, being a finite set of points and a set of subsets of called blocks satisfying the following simple property for all : if , then . To generalize results on the average order of a connected induced subgraph of a graph, it is conjectured that if a UIS is, in various senses, “connected and regular,” then the average size of a block is at least half the number of points. We prove that if a union‐intersection set system is regular, completely irreducible, and nonredundant, then the average size of a block is at least half the number of points.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43886766","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":"Properly colored and rainbow \u0000 \u0000 \u0000 \u0000 C\u0000 4\u0000 \u0000 \u0000 ${C}_{4}$\u0000 's in edge-colored graphs","authors":"Fangfang Wu,&nbsp;Hajo Broersma,&nbsp;Shenggui Zhang,&nbsp;Binlong Li","doi":"10.1002/jgt.23019","DOIUrl":"10.1002/jgt.23019","url":null,"abstract":"<p>We present new sharp sufficient conditions for the existence of properly colored and rainbow <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${C}_{4}$</annotation>\u0000 </semantics></math>'s in edge-colored graphs. Our first results deal with sharp color neighborhood conditions for the existence of properly colored <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${C}_{4}$</annotation>\u0000 </semantics></math>'s in edge-colored complete graphs and complete bipartite graphs, respectively. Next, we characterize the extremal graphs for an anti-Ramsey number result due to Alon on the existence of rainbow <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${C}_{4}$</annotation>\u0000 </semantics></math>'s in edge-colored complete graphs. We also generalize Alon's result from complete to general edge-colored graphs. Finally, we derive a structural property regarding the extremal graphs for a bipartite counterpart of Alon's result due to Axenovich, Jiang, and Kündgen on the existence of rainbow <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${C}_{4}$</annotation>\u0000 </semantics></math>'s in edge-colored complete bipartite graphs. We also generalize their result from complete to general bipartite edge-colored graphs.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"105 1","pages":"110-135"},"PeriodicalIF":0.9,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48011581","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 conditions implying stability of graphs","authors":"Ademir Hujdurović,&nbsp;Ðorđe Mitrović","doi":"10.1002/jgt.23018","DOIUrl":"10.1002/jgt.23018","url":null,"abstract":"<p>A graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math> is said to be <i>unstable</i> if the direct product <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $Xtimes {K}_{2}$</annotation>\u0000 </semantics></math> (also called the <i>canonical double cover</i> of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math>) has automorphisms that do not come from automorphisms of its factors <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${K}_{2}$</annotation>\u0000 </semantics></math>. It is <i>nontrivially unstable</i> if it is unstable, connected, nonbipartite, and distinct vertices have distinct sets of neighbours. In this paper, we prove two sufficient conditions for stability of graphs in which every edge lies on a triangle, revising an incorrect claim of Surowski and filling in some gaps in the proof of another one. We also consider triangle-free graphs, and prove that there are no nontrivially unstable triangle-free graphs of diameter 2. An interesting construction of nontrivially unstable graphs is given and several open problems are posed.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"105 1","pages":"98-109"},"PeriodicalIF":0.9,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41825876","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":"Subdivisions with congruence constraints in digraphs of large chromatic number","authors":"Raphael Steiner","doi":"10.1002/jgt.23020","DOIUrl":"10.1002/jgt.23020","url":null,"abstract":"&lt;p&gt;We prove that for every digraph &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $F$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and every assignment of pairs of integers &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${({r}_{e},{q}_{e})}_{ein A(F)}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; to its arcs there exists an integer &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that every digraph &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $D$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with dichromatic number greater than &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; contains a subdivision of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $F$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in which &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $e$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is subdivided into a directed path ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"105 1","pages":"136-143"},"PeriodicalIF":0.9,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50788894","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":"Turán numbers \u0000 \u0000 \u0000 T\u0000 \u0000 (\u0000 \u0000 n\u0000 ,\u0000 5\u0000 ,\u0000 3\u0000 \u0000 )\u0000 \u0000 \u0000 $T(n,5,3)$\u0000 and graphs without induced 5-cycles","authors":"Iliya Bluskov,&nbsp;Jan de Heer,&nbsp;Alexander Sidorenko","doi":"10.1002/jgt.23021","DOIUrl":"10.1002/jgt.23021","url":null,"abstract":"&lt;p&gt;The Turán number &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $T(n,5,3)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the minimum size of a system of triples out of a base set &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; elements such that every quintuple in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; contains a triple from the system. The exact values of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $T(n,5,3)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; are known for &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≤&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;17&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $nle 17$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Turán conjectured that &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 \u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"104 3","pages":"451-460"},"PeriodicalIF":0.9,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45594879","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":"Random polynomial graphs for random Turán problems","authors":"Sam Spiro","doi":"10.1002/jgt.23015","DOIUrl":"10.1002/jgt.23015","url":null,"abstract":"&lt;p&gt;Bukh and Conlon used random polynomial graphs to give effective lower bounds on &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;ex&lt;/mtext&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $text{ex}(n,{{mathscr{T}}}^{ell })$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${{mathscr{T}}}^{ell }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $ell $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;th power of a balanced rooted tree &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $T$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We extend their result to give effective lower bounds on &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;ex&lt;/mtext&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $text{ex}({G}_{n,p},{{mathscr{T}}}^{ell })$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, which is the maximum number of edges in a &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${{mathscr{T}}}^{ell }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-free subgraph of the random graph &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"105 2","pages":"192-208"},"PeriodicalIF":0.9,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47905355","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":"Maximally edge-connected realizations and Kundu's \u0000 \u0000 \u0000 k\u0000 \u0000 $k$\u0000 -factor theorem","authors":"James M. Shook","doi":"10.1002/jgt.23017","DOIUrl":"10.1002/jgt.23017","url":null,"abstract":"&lt;p&gt;A simple graph &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with edge-connectivity &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $lambda (G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and minimum degree &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;δ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $delta (G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is maximally edge-connected if &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;δ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $lambda (G)=delta (G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In 1964, given a nonincreasing degree sequence &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;π&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;…&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $pi =({d}_{1},{rm{ldots }},{d}_{n})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, Jack Edmonds showed that there is a realization &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;math&gt;\u0000","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"105 1","pages":"83-97"},"PeriodicalIF":0.9,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46967648","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}