Bipartite Ramsey number pairs that involve combinations of cycles and odd paths

IF 0.7 3区 数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2024-10-15 DOI:10.1016/j.disc.2024.114283
Ernst J. Joubert
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Joubert","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":"348 2","pages":"Article 114283"},"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 G1,G2,,Gk, the bipartite Ramsey number b(G1,G2, ,Gk) is the least positive integer b, so that any coloring of the edges of Kb,b with k colors, will result in a copy of Gi in the ith color, for some i. For bipartite graphs G1 and G2, the bipartite Ramsey number pair (a,b), denoted by bp(G1,G2)=(a,b), is an ordered pair of integers such that for any blue-red coloring of the edges of Ka,b, with ab, either a blue copy of G1 exists or a red copy of G2 exists if and only if aa and bb. In [4], Faudree and Schelp considered bipartite Ramsey number pairs involving paths. Recently, Joubert, Hattingh and Henning showed, in [7] and [8], that bp(C2s,C2s)=(2s,2s1) and b(P2s,C2s)=2s1, 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 bp(C2s,P2r+1)=(s+r,s+r1) if rs+1, bp(P2s+1,C2r)=(s+r,s+r) if r=s+1, and bp(P2s+1,C2r)=(s+r1,s+r1) if rs+2.
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涉及循环和奇数路径组合的二方拉姆齐数对
对于双胞图 G1,G2,...,Gk,双胞拉姆齐数 b(G1,G2,...,Gk)是最小的正整数 b,使得 Kb,b 的边的任何着色都有 k 种颜色,在第 i 种颜色下将得到 Gi 的副本。对于双分部图 G1 和 G2,双分部拉姆齐数对(a,b)(用 bp(G1,G2)=(a,b) 表示)是一对有序整数,对于 Ka′、b′,当且仅当 a′≥a,b′≥b 时,要么存在 G1 的蓝色副本,要么存在 G2 的红色副本。在 [4] 中,Faudree 和 Schelp 考虑了涉及路径的双方位拉姆齐数对。最近,Joubert、Hattingh 和 Henning 在 [7] 和 [8] 中证明,对于足够大的正整数 s,bp(C2s,C2s)=(2s,2s-1) 和 b(P2s,C2s)=2s-1。具体来说,假设 s 和 r 是足够大的正整数。我们将证明,如果 r≥s+1 时,bp(C2s,P2r+1)=(s+r,s+r-1);如果 r=s+1 时,bp(P2s+1,C2r)=(s+r,s+r);如果 r≥s+2 时,bp(P2s+1,C2r)=(s+r-1,s+r-1)。
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来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
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. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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