Journal of Graph Theory最新文献

{"title":"Arc-disjoint out-branchings and in-branchings in semicomplete digraphs","authors":"J. Bang-Jensen,&nbsp;Y. Wang","doi":"10.1002/jgt.23072","DOIUrl":"10.1002/jgt.23072","url":null,"abstract":"<p>An out-branching <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>B</mi>\u0000 \u0000 <mi>u</mi>\u0000 \u0000 <mo>+</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation> ${B}_{u}^{+}$</annotation>\u0000 </semantics></math> (in-branching <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>B</mi>\u0000 \u0000 <mi>u</mi>\u0000 \u0000 <mo>−</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation> ${B}_{u}^{-}$</annotation>\u0000 </semantics></math>) in a digraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> is a connected spanning subdigraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> in which every vertex except the vertex <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>u</mi>\u0000 </mrow>\u0000 <annotation> $u$</annotation>\u0000 </semantics></math>, called the root, has in-degree (out-degree) one. It is well known that there exists a polynomial algorithm for deciding whether a given digraph has <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> arc-disjoint out-branchings with prescribed roots (<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> is part of the input). In sharp contrast to this, it is already NP-complete to decide if a digraph has one out-branching which is arc-disjoint from some in-branching. A digraph is <i>semicomplete</i> if it has no pair of nonadjacent vertices. A <i>tournament</i> is a semicomplete digraph without directed cycles of length 2. In this paper we give a complete classification of semicomplete digraphs that have an out-branching <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>B</mi>\u0000 \u0000 <mi>u</mi>\u0000 \u0000 <mo>+</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation> ${B}_{u}^{+}$</annotation>\u0000 </semantics></math> which is arc-disjoint from some in-branching <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>B</mi>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139093661","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":"New eigenvalue bound for the fractional chromatic number","authors":"Krystal Guo,&nbsp;Sam Spiro","doi":"10.1002/jgt.23071","DOIUrl":"10.1002/jgt.23071","url":null,"abstract":"<p>Given a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, we let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>s</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${s}^{+}(G)$</annotation>\u0000 </semantics></math> denote the sum of the squares of the positive eigenvalues of the adjacency matrix of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, and we similarly define <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>s</mi>\u0000 <mo>−</mo>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${s}^{-}(G)$</annotation>\u0000 </semantics></math>. We prove that\u0000\u0000 </p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23071","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139056381","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":"Optimal linear-Vizing relationships for (total) domination in graphs","authors":"Michael A. Henning,&nbsp;Paul Horn","doi":"10.1002/jgt.23070","DOIUrl":"10.1002/jgt.23070","url":null,"abstract":"<p>A total dominating set in a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a set of vertices of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> such that every vertex is adjacent to a vertex of the set. The total domination number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>γ</mi>\u0000 \u0000 <mi>t</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${gamma }_{t}(G)$</annotation>\u0000 </semantics></math> is the minimum cardinality of a total dominating set in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. In this paper, we study the following open problem posed by Yeo. For each <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation> ${rm{Delta }}ge 3$</annotation>\u0000 </semantics></math>, find the smallest value, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>r</mi>\u0000 \u0000 <mi>Δ</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${r}_{{rm{Delta }}}$</annotation>\u0000 </semantics></math>, such that every connected graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> of order at least 3, of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>, size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>, total domination number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>γ</mi>\u0000 \u0000 <mi>t</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${gamma }_{t}$</annotation>\u0000 </semantics></math>, and bounded maximum degree <math>\u0000 <semantics>\u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138826097","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":"Classes of intersection digraphs with good algorithmic properties","authors":"Lars Jaffke,&nbsp;O-joung Kwon,&nbsp;Jan Arne Telle","doi":"10.1002/jgt.23065","DOIUrl":"10.1002/jgt.23065","url":null,"abstract":"<p>While intersection graphs play a central role in the algorithmic analysis of hard problems on undirected graphs, the role of intersection <i>digraphs</i> in algorithms is much less understood. We present several contributions towards a better understanding of the algorithmic treatment of intersection digraphs. First, we introduce natural classes of intersection digraphs that generalize several classes studied in the literature. Second, we define the directed locally checkable vertex (DLCV) problems, which capture many well-studied problems on digraphs, such as \u0000<span>(Independent) Dominating Set</span>, \u0000<span>Kernel</span>, and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math>-<span>Homomorphism.</span> Third, we give a new width measure of digraphs, <i>bi-mim-width</i>, and show that the DLCV problems are polynomial-time solvable when we are provided a decomposition of small bi-mim-width. Fourth, we show that several classes of intersection digraphs have bounded bi-mim-width, implying that we can solve all DLCV problems on these classes in polynomial time given an intersection representation of the input digraph. We identify reflexivity as a useful condition to obtain intersection digraph classes of bounded bi-mim-width, and therefore to obtain positive algorithmic results.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23065","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138715321","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":"Exact values for some unbalanced Zarankiewicz numbers","authors":"Guangzhou Chen,&nbsp;Daniel Horsley,&nbsp;Adam Mammoliti","doi":"10.1002/jgt.23068","DOIUrl":"10.1002/jgt.23068","url":null,"abstract":"<p>For positive integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 <annotation> $s$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>, the Zarankiewicz number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>t</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${Z}_{s,t}(m,n)$</annotation>\u0000 </semantics></math> is defined to be the maximum number of edges in a bipartite graph with parts of sizes <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> that has no complete bipartite subgraph containing <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 <annotation> $s$</annotation>\u0000 </semantics></math> vertices in the part of size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math> vertices in the part of size <math>\u0000 <semantics>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23068","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138679930","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":"Overfullness of edge-critical graphs with small minimal core degree","authors":"Yan Cao,&nbsp;Guantao Chen,&nbsp;Guangming Jing,&nbsp;Songling Shan","doi":"10.1002/jgt.23069","DOIUrl":"10.1002/jgt.23069","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> be a simple graph. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Delta }}(G)$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 \u0000 <mo>′</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $chi ^{prime} (G)$</annotation>\u0000 </semantics></math> be the maximum degree and the chromatic index of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, respectively. We call <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> <i>overfull</i> if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>E</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>∕</mo>\u0000 \u0000 <mrow>\u0000 <mo>⌊</mo>\u0000 \u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>∕</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>⌋</mo>\u0000 </mrow>\u0000 \u0000 <mo>&gt;</mo>\u0000 \u0000 <mi>Δ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138679918","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":"Rainbow subgraphs in edge-colored complete graphs: Answering two questions by Erdős and Tuza","authors":"Maria Axenovich,&nbsp;Felix C. Clemen","doi":"10.1002/jgt.23063","DOIUrl":"10.1002/jgt.23063","url":null,"abstract":"<p>An edge-coloring of a complete graph with a set of colors <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> $C$</annotation>\u0000 </semantics></math> is called <i>completely balanced</i> if any vertex is incident to the same number of edges of each color from <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> $C$</annotation>\u0000 </semantics></math>. Erdős and Tuza asked in 1993 whether for any graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow>\u0000 <annotation> $ell $</annotation>\u0000 </semantics></math> edges and any completely balanced coloring of any sufficiently large complete graph using <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow>\u0000 <annotation> $ell $</annotation>\u0000 </semantics></math> colors contains a rainbow copy of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math>. This question was restated by Erdős in his list of “Some of my favourite problems on cycles and colourings.” We answer this question in the negative for most cliques <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>K</mi>\u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $F={K}_{q}$</annotation>\u0000 </semantics></math> by giving explicit constructions of respective completely balanced colorings. Further, we answer a related question concerning completely balanced colorings of complete graphs with more colors than the number of edges in the graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23063","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138576573","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":"Partitioning kite-free planar graphs into two forests","authors":"Yang Wang,&nbsp;Yiqiao Wang,&nbsp;Ko-Wei Lih","doi":"10.1002/jgt.23062","DOIUrl":"10.1002/jgt.23062","url":null,"abstract":"<p>A kite is a complete graph on four vertices with one edge removed. It is proved that every planar graph without a kite as subgraph can be partitioned into two induced forests. This resolves a conjecture of Raspaud and Wang in 2008.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138576182","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 Turán problems with bounded matching number","authors":"Dániel Gerbner","doi":"10.1002/jgt.23067","DOIUrl":"10.1002/jgt.23067","url":null,"abstract":"<p>Very recently, Alon and Frankl initiated the study of the maximum number of edges in <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>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math>-free graphs with matching number at most <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 <annotation> $s$</annotation>\u0000 </semantics></math>. For fixed <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 <annotation> $s$</annotation>\u0000 </semantics></math>, we determine this number apart from a constant additive term. We also obtain several exact results.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138545944","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":"5-Coloring reconfiguration of planar graphs with no short odd cycles","authors":"Daniel W. Cranston,&nbsp;Reem Mahmoud","doi":"10.1002/jgt.23064","DOIUrl":"10.1002/jgt.23064","url":null,"abstract":"<p>The coloring reconfiguration graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{mathscr{C}}}_{k}(G)$</annotation>\u0000 </semantics></math> has as its vertex set all the proper <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-colorings of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, and two vertices in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{mathscr{C}}}_{k}(G)$</annotation>\u0000 </semantics></math> are adjacent if their corresponding <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-colorings differ on a single vertex. Cereceda conjectured that if an <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math>-degenerate and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mi>d</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $kge d+2$</annotation>\u0000 </semantics></math>, then the diameter of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138546072","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}