Journal of Graph Theory最新文献

{"title":"Five‐cycle double cover and shortest cycle cover","authors":"Siyan Liu, Rong‐Xia Hao, Rong Luo, Cun‐Quan Zhang","doi":"10.1002/jgt.23164","DOIUrl":"https://doi.org/10.1002/jgt.23164","url":null,"abstract":"The 5‐even subgraph cycle double cover conjecture (5‐CDC conjecture) asserts that every bridgeless graph has a 5‐even subgraph double cover. A shortest even subgraph cover of a graph is a family of even subgraphs which cover all the edges of and the sum of their lengths is minimum. It is conjectured that every bridgeless graph has an even subgraph cover with total length at most . In this paper, we study those two conjectures for weak oddness 2 cubic graphs and present a sufficient condition for such graphs to have a 5‐CDC containing a member with many vertices. As a corollary, we show that for every oddness 2 cubic graph satisfying the sufficient condition has a 4‐even subgraph ‐cover with total length at most . We also show that every oddness 2 cubic graph with girth at least 30 has a 5‐CDC containing a member of length at least and thus it has a 4‐even subgraph ‐cover with total length at most .","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943300","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":"Dense circuit graphs and the planar Turán number of a cycle","authors":"Ruilin Shi, Zach Walsh, Xingxing Yu","doi":"10.1002/jgt.23165","DOIUrl":"https://doi.org/10.1002/jgt.23165","url":null,"abstract":"The <jats:italic>planar Turán number</jats:italic> of a graph is the maximum number of edges in an ‐vertex planar graph without as a subgraph. Let denote the cycle of length . The planar Turán number is known for . We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Turán numbers. In particular, we prove that there is a constant so that for all and . When this bound is tight up to the constant and proves a conjecture of Cranston, Lidický, Liu, and Shantanam.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943337","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":"Strong arc decompositions of split digraphs","authors":"Jørgen Bang‐Jensen, Yun Wang","doi":"10.1002/jgt.23157","DOIUrl":"https://doi.org/10.1002/jgt.23157","url":null,"abstract":"A <jats:italic>strong arc decomposition</jats:italic> of a digraph is a partition of its arc set into two sets such that the digraph is strong for . Bang‐Jensen and Yeo conjectured that there is some such that every ‐arc‐strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2‐arc‐strong semicomplete digraph has a strong arc decomposition. Bang‐Jensen and Huang extended this result to locally semicomplete digraphs by proving that every 2‐arc‐strong locally semicomplete digraph which is not the square of an even cycle has a strong arc decomposition. This implies that every 3‐arc‐strong locally semicomplete digraph has a strong arc decomposition. A <jats:italic>split digraph</jats:italic> is a digraph whose underlying undirected graph is a split graph, meaning that its vertices can be partitioned into a clique and an independent set. Equivalently, a split digraph is any digraph which can be obtained from a semicomplete digraph by adding a new set of vertices and some arcs between and . In this paper, we prove that every 3‐arc‐strong split digraph has a strong arc decomposition which can be found in polynomial time and we provide infinite classes of 2‐strong split digraphs with no strong arc decomposition. We also pose a number of open problems on split digraphs.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943301","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":"Stability of Rose Window graphs","authors":"Milad Ahanjideh,&nbsp;István Kovács,&nbsp;Klavdija Kutnar","doi":"10.1002/jgt.23162","DOIUrl":"10.1002/jgt.23162","url":null,"abstract":"<p>A graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}$</annotation>\u0000 </semantics></math> is said to be stable if for the direct product <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mtext>Aut</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}times {{bf{K}}}_{2},text{Aut}({rm{Gamma }}times {{bf{K}}}_{2})$</annotation>\u0000 </semantics></math> is isomorphic to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Aut</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>Γ</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>Z</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{Aut}({rm{Gamma }})times {{mathbb{Z}}}_{2}$</annotation>\u0000 </semantics></math>; otherwise, it is called unstable. An unstable graph is called nontrivially unstable when it is not bipartite and no two vertices have the same neighborhood. Wilson described nine families of unstable Rose Window graphs and conjectured that these contain all nontrivially unstable Rose Window graphs (2008). In this paper we show that the conjecture is true.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943338","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":"Proper edge colorings of planar graphs with rainbow \u0000 \u0000 \u0000 \u0000 \u0000 C\u0000 4\u0000 \u0000 \u0000 \u0000 ${C}_{4}$\u0000 -s","authors":"András Gyárfás,&nbsp;Ryan R. Martin,&nbsp;Miklós Ruszinkó,&nbsp;Gábor N. Sárközy","doi":"10.1002/jgt.23163","DOIUrl":"10.1002/jgt.23163","url":null,"abstract":"&lt;p&gt;We call a proper edge coloring of a graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; a B-coloring if every 4-cycle of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is colored with four different colors. Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 \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;/mrow&gt;\u0000 &lt;annotation&gt; ${q}_{B}(G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; denote the smallest number of colors needed for a B-coloring of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Motivated by earlier papers on B-colorings, here we consider &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 \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;/mrow&gt;\u0000 &lt;annotation&gt; ${q}_{B}(G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for planar and outerplanar graphs in terms of the maximum degree &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 \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;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{Delta }}={rm{Delta }}(G)$&lt;/annotation&gt;\u0000 &lt;/sem","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943351","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":"Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs”","authors":"","doi":"10.1002/jgt.23160","DOIUrl":"10.1002/jgt.23160","url":null,"abstract":"<p>A. Espuny Díaz, L. Lichev, D. Mitsche, and A. Wesolek, <i>Sharp threshold for embedding balanced spanning trees in random geometric graphs</i>, J. Graph Theory. <b>107</b> (2024), 107–125. https://doi.org/10.1002/jgt.23106</p><p>In the “ACKNOWLEDGMENTS” section, the text “The research leading to these results has been supported by the Carl-Zeiss-Foundation (Alberto Espuny Díaz), by grant GrHyDy ANR-20-CE40-0002, and by Fondecyt grant 1220174 (Dieter Mitsche) and by the Vanier Scholarship Program (Alexandra Wesolek).” was incorrect.</p><p>This should have read: “The research leading to these results has been supported by the Carl-Zeiss-Foundation and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through project no. 447645533 (A. Espuny Díaz), by grant GrHyDy ANR-20-CE40-0002 and by Fondecyt grant 1220174 (D. Mitsche) and by the Vanier Scholarship Program (A. Wesolek).”</p><p>The new Funding Information should read as follows:</p><p>Carl-Zeiss-Foundation.</p><p>Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project no. 447645533.</p><p>Grant GrHyDy ANR-20-CE40-0002.</p><p>Fondecyt, Grant/Award Number: 1220174.</p><p>The Vanier Scholarship Program.</p><p>We apologize for this error.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23160","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866312","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":"On the minimum number of arcs in 4-dicritical oriented graphs","authors":"Frédéric Havet,&nbsp;Lucas Picasarri-Arrieta,&nbsp;Clément Rambaud","doi":"10.1002/jgt.23159","DOIUrl":"10.1002/jgt.23159","url":null,"abstract":"&lt;p&gt;The dichromatic number &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $overrightarrow{chi }(D)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of a digraph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $D$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $D$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dicritical if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;D&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;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $overrightarrow{chi }(D)=k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and each proper subdigraph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $H$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $D$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; satisfies &lt;span&gt;&lt;/span&gt;&lt;mat","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866313","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":"Counting triangles in regular graphs","authors":"Jialin He,&nbsp;Xinmin Hou,&nbsp;Jie Ma,&nbsp;Tianying Xie","doi":"10.1002/jgt.23156","DOIUrl":"10.1002/jgt.23156","url":null,"abstract":"&lt;p&gt;In this paper, we investigate the minimum number of triangles, denoted by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $t(n,k)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-vertex &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-regular graphs, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is an odd integer and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is an even integer. The well-known Andrásfai–Erdős–Sós Theorem has established that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;an","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23156","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771226","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":"Counting rainbow triangles in edge-colored graphs","authors":"Xueliang Li,&nbsp;Bo Ning,&nbsp;Yongtang Shi,&nbsp;Shenggui Zhang","doi":"10.1002/jgt.23158","DOIUrl":"10.1002/jgt.23158","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be an edge-colored graph on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; vertices. The minimum color degree of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, denoted by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;δ&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 \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;/mrow&gt;\u0000 &lt;annotation&gt; ${delta }^{c}(G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, is defined as the minimum number of colors assigned to the edges incident to a vertex in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In 2013, Li proved that an edge-colored graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; vertices contains a rainbow triangle if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;δ&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 \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;/m","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771227","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":"Defective acyclic colorings of planar graphs","authors":"On-Hei Solomon Lo,&nbsp;Ben Seamone,&nbsp;Xuding Zhu","doi":"10.1002/jgt.23154","DOIUrl":"10.1002/jgt.23154","url":null,"abstract":"&lt;p&gt;This paper studies two variants of defective acyclic coloring of planar graphs. For a graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and a coloring &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;φ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $varphi $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, a 2-colored cycle (2CC) transversal is a subset &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${E}^{^{prime} }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 \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;/mrow&gt;\u0000 &lt;annotation&gt; $E(G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; that intersects every 2-colored cycle. Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a positive integer. We denote by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 \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;/mrow&gt;\u0000 &lt;annotation&gt; ${m}_{k}(G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; the minimum integer &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744614","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}