{"title":"Bipartite Ramsey number pairs that involve combinations of cycles and odd paths","authors":"","doi":"10.1016/j.disc.2024.114283","DOIUrl":null,"url":null,"abstract":"<div><div>For bipartite graphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, the bipartite Ramsey number <span><math><mi>b</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> is the least positive integer <em>b</em>, so that any coloring of the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>b</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> with <em>k</em> colors, will result in a copy of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> in the <em>i</em>th color, for some <em>i</em>. For bipartite graphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, the bipartite Ramsey number pair <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>, denoted by <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>, is an ordered pair of integers such that for any blue-red coloring of the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span>, with <span><math><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, either a blue copy of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> exists or a red copy of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> exists if and only if <span><math><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><mi>a</mi></math></span> and <span><math><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><mi>b</mi></math></span>. In <span><span>[4]</span></span>, Faudree and Schelp considered bipartite Ramsey number pairs involving paths. Recently, Joubert, Hattingh and Henning showed, in <span><span>[7]</span></span> and <span><span>[8]</span></span>, that <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mn>2</mn><mi>s</mi><mo>,</mo><mn>2</mn><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mi>b</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>2</mn><mi>s</mi><mo>−</mo><mn>1</mn></math></span>, for sufficiently large positive integers <em>s</em>. In this paper we will focus our attention on finding exact values for bipartite Ramsey number pairs that involve cycles and odd paths. Specifically, let <em>s</em> and <em>r</em> be sufficiently large positive integers. We will prove that <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mi>s</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> if <span><math><mi>r</mi><mo>≥</mo><mi>s</mi><mo>+</mo><mn>1</mn></math></span>, <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mi>s</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>+</mo><mi>r</mi><mo>)</mo></math></span> if <span><math><mi>r</mi><mo>=</mo><mi>s</mi><mo>+</mo><mn>1</mn></math></span>, and <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mi>s</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> if <span><math><mi>r</mi><mo>≥</mo><mi>s</mi><mo>+</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X2400414X","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
For bipartite graphs , the bipartite Ramsey number , is the least positive integer b, so that any coloring of the edges of with k colors, will result in a copy of in the ith color, for some i. For bipartite graphs and , the bipartite Ramsey number pair , denoted by , is an ordered pair of integers such that for any blue-red coloring of the edges of , with , either a blue copy of exists or a red copy of exists if and only if and . In [4], Faudree and Schelp considered bipartite Ramsey number pairs involving paths. Recently, Joubert, Hattingh and Henning showed, in [7] and [8], that and , for sufficiently large positive integers s. In this paper we will focus our attention on finding exact values for bipartite Ramsey number pairs that involve cycles and odd paths. Specifically, let s and r be sufficiently large positive integers. We will prove that if , if , and if .
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
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