{"title":"5-Coloring reconfiguration of planar graphs with no short odd cycles","authors":"Daniel W. Cranston, Reem Mahmoud","doi":"10.1002/jgt.23064","DOIUrl":null,"url":null,"abstract":"<p>The coloring reconfiguration graph <math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n \n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${{\\mathscr{C}}}_{k}(G)$</annotation>\n </semantics></math> has as its vertex set all the proper <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-colorings of <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, and two vertices in <math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n \n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${{\\mathscr{C}}}_{k}(G)$</annotation>\n </semantics></math> are adjacent if their corresponding <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-colorings differ on a single vertex. Cereceda conjectured that if an <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>-vertex graph <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math>-degenerate and <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mi>d</mi>\n \n <mo>+</mo>\n \n <mn>2</mn>\n </mrow>\n <annotation> $k\\ge d+2$</annotation>\n </semantics></math>, then the diameter of <math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n \n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${{\\mathscr{C}}}_{k}(G)$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n <mi>O</mi>\n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>n</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $O({n}^{2})$</annotation>\n </semantics></math>. Bousquet and Heinrich proved that if <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is planar and bipartite, then the diameter of <math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n \n <mn>5</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${{\\mathscr{C}}}_{5}(G)$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n <mi>O</mi>\n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>n</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $O({n}^{2})$</annotation>\n </semantics></math>. (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is planar and has no 3-cycles. As a partial solution to this problem, we show that the diameter of <math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n \n <mn>5</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${{\\mathscr{C}}}_{5}(G)$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n <mi>O</mi>\n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>n</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $O({n}^{2})$</annotation>\n </semantics></math> for every planar graph <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with no 3-cycles and no 5-cycles.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23064","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The coloring reconfiguration graph has as its vertex set all the proper -colorings of , and two vertices in are adjacent if their corresponding -colorings differ on a single vertex. Cereceda conjectured that if an -vertex graph is -degenerate and , then the diameter of is . Bousquet and Heinrich proved that if is planar and bipartite, then the diameter of is . (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when is planar and has no 3-cycles. As a partial solution to this problem, we show that the diameter of is for every planar graph with no 3-cycles and no 5-cycles.
期刊介绍:
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
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