We restate Thomassen's theorem of 3-extendability (Thomassen, Journal of Combinatorial Theory Series B, 97, 571–583), an extension of the famous planar 5-choosability theorem, in terms of graph polynomials. This yields an Alon–Tarsi equivalent of 3-extendability.
我们用图多项式的形式重申了Thomassen的3-可扩展性定理（Thomassen, Journal of Combinatorial Theory Series B, 97, 571-583），这是著名的平面5-可选择性定理的推广。这产生了相当于3-可扩展性的Alon-Tarsi。
{"title":"The Polynomial Method for Three-Path Extendability of List Colourings of Planar Graphs","authors":"Przemysław Gordinowicz, Paweł Twardowski","doi":"10.1002/jgt.23214","DOIUrl":"https://doi.org/10.1002/jgt.23214","url":null,"abstract":"<div>\u0000 \u0000 <p>We restate Thomassen's theorem of 3-extendability (Thomassen, Journal of Combinatorial Theory Series B, 97, 571–583), an extension of the famous planar 5-choosability theorem, in terms of graph polynomials. This yields an Alon–Tarsi equivalent of 3-extendability.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 2","pages":"237-254"},"PeriodicalIF":0.9,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143846078","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}
A subgraph of a multigraph is overfull if Analogous to the Overfull Conjecture proposed by Chetwynd and Hilton in 1986, Stiebitz et al. formed the multigraph version of the conjecture as follows: Let be a multigraph with maximum multiplicity and maximum degree
Agnijo Banerjee, João Pedro Marciano, Adva Mond, Jan Petr, Julien Portier
<div> <p>Deciding whether a graph can be edge-decomposed into a matching and a <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> <annotation> <math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0001" wiley:location="equation/jgt23208-math-0001.png"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation> </semantics></math>-bounded linear forest was recently shown by Campbell, Hörsch, and Moore to be nonedeterministic Polynomial time (NP)-complete for every <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> <mo>≥</mo> <mn>9</mn> </mrow> </mrow> <annotation> <math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0002" wiley:location="equation/jgt23208-math-0002.png"><mrow><mrow><mi>k</mi><mo>unicode{x02265}</mo><mn>9</mn></mrow></mrow></math></annotation> </semantics></math>, and solvable in polynomial time for <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </mrow> <annotation> <math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0003" wiley:location="equation/jgt23208-math-0003.png"><mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math></annotation> </semantics></math>. In the first part of this paper, we close this gap by showing that this problem is NP-complete for every <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </mrow> <annotation> <math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0004" wiley:location="equation/jgt23208-math-0004.png"><mrow><mrow><mi>k</mi><mo>unicode{x02265}</mo><mn>3</mn>
{"title":"The Complexity of Decomposing a Graph into a Matching and a Bounded Linear Forest","authors":"Agnijo Banerjee, João Pedro Marciano, Adva Mond, Jan Petr, Julien Portier","doi":"10.1002/jgt.23208","DOIUrl":"https://doi.org/10.1002/jgt.23208","url":null,"abstract":"<div>\u0000 \u0000 <p>Deciding whether a graph can be edge-decomposed into a matching and a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0001\" wiley:location=\"equation/jgt23208-math-0001.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-bounded linear forest was recently shown by Campbell, Hörsch, and Moore to be nonedeterministic Polynomial time (NP)-complete for every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>9</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0002\" wiley:location=\"equation/jgt23208-math-0002.png\"><mrow><mrow><mi>k</mi><mo>unicode{x02265}</mo><mn>9</mn></mrow></mrow></math></annotation>\u0000 </semantics></math>, and solvable in polynomial time for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0003\" wiley:location=\"equation/jgt23208-math-0003.png\"><mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math></annotation>\u0000 </semantics></math>. In the first part of this paper, we close this gap by showing that this problem is NP-complete for every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0004\" wiley:location=\"equation/jgt23208-math-0004.png\"><mrow><mrow><mi>k</mi><mo>unicode{x02265}</mo><mn>3</mn>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"76-87"},"PeriodicalIF":0.9,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612438","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}
We determine the maximum number of copies of in a -free -vertex graph for all integers and sufficiently large . Moreover, for and any integer , we obtain the maximum number of cycles of length in an -vertex -free bipartite graph.
In this paper, we study the maximum degree in nonempty-induced subgraphs of the Kneser graph . One of the main results asserts that, for and , whenever a nonempty subgraph has
A digraph is called ubiquitous if every digraph that contains arbitrarily many vertex-disjoint copies of also contains infinitely many vertex-disjoint copies of . We study oriented double rays, that is, digraphs whose underlying undirected graphs are double rays. Calling a vertex of an oriented double ray a turn if it has in-degree or out-degree 2, we prove that an oriented double ray with at least one turn is ubiquitous if and only if it has a (finite) odd number of turns. It remains an open problem to determine whether the consistently oriented double ray is ubiquitous.
{"title":"On the Ubiquity of Oriented Double Rays","authors":"Florian Gut, Thilo Krill, Florian Reich","doi":"10.1002/jgt.23216","DOIUrl":"https://doi.org/10.1002/jgt.23216","url":null,"abstract":"<p>A digraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23216:jgt23216-math-0001\" wiley:location=\"equation/jgt23216-math-0001.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> is called <i>ubiquitous</i> if every digraph that contains arbitrarily many vertex-disjoint copies of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23216:jgt23216-math-0002\" wiley:location=\"equation/jgt23216-math-0002.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> also contains infinitely many vertex-disjoint copies of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23216:jgt23216-math-0003\" wiley:location=\"equation/jgt23216-math-0003.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>. We study oriented double rays, that is, digraphs <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23216:jgt23216-math-0004\" wiley:location=\"equation/jgt23216-math-0004.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> whose underlying undirected graphs are double rays. Calling a vertex of an oriented double ray a turn if it has in-degree or out-degree 2, we prove that an oriented double ray with at least one turn is ubiquitous if and only if it has a (finite) odd number of turns. It remains an open problem to determine whether the consistently oriented double ray is ubiquitous.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"62-67"},"PeriodicalIF":0.9,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23216","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612347","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}
Applications of graph colouring often involve taking restrictions into account, and it is desirable to have multiple (disjoint) solutions. In the optimal case, where there is a partition into disjoint colourings, we speak of a packing. However, even for complete bipartite graphs, the list chromatic number can be arbitrarily large, and its exact determination is generally difficult. For the packing variant, this question becomes even harder. In this paper, we study the correspondence- and list-packing numbers of (asymmetric) complete bipartite graphs. In the most asymmetric cases, Latin squares come into play. Our results show that every can be equal to the correspondence packing number of a graph. We disprove a recent conjecture that relates the list packing number and the list flexibility number. Additionally, we improve the threshold functions for the correspondence packing variant.
{"title":"Packing Colourings in Complete Bipartite Graphs and the Inverse Problem for Correspondence Packing","authors":"Stijn Cambie, Rimma Hämäläinen","doi":"10.1002/jgt.23215","DOIUrl":"https://doi.org/10.1002/jgt.23215","url":null,"abstract":"<div>\u0000 \u0000 <p>Applications of graph colouring often involve taking restrictions into account, and it is desirable to have multiple (disjoint) solutions. In the optimal case, where there is a partition into disjoint colourings, we speak of a packing. However, even for complete bipartite graphs, the list chromatic number can be arbitrarily large, and its exact determination is generally difficult. For the packing variant, this question becomes even harder. In this paper, we study the correspondence- and list-packing numbers of (asymmetric) complete bipartite graphs. In the most asymmetric cases, Latin squares come into play. Our results show that every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>z</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <msup>\u0000 <mi>Z</mi>\u0000 \u0000 <mo>+</mo>\u0000 </msup>\u0000 \u0000 <mo></mo>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23215:jgt23215-math-0001\" wiley:location=\"equation/jgt23215-math-0001.png\"><mrow><mrow><mi>z</mi><mo>unicode{x02208}</mo><msup><mi mathvariant=\"double-struck\">Z</mi><mo>unicode{x0002B}</mo></msup><mo>unicode{x0005C}</mo><mrow><mo class=\"MathClass-open\">{</mo><mn>3</mn><mo class=\"MathClass-close\">}</mo></mrow></mrow></mrow></math></annotation>\u0000 </semantics></math> can be equal to the correspondence packing number of a graph. We disprove a recent conjecture that relates the list packing number and the list flexibility number. Additionally, we improve the threshold functions for the correspondence packing variant.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"52-61"},"PeriodicalIF":0.9,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612344","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}
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