{"title":"涉及循环和奇数路径组合的二方拉姆齐数对","authors":"Ernst J. 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":"{\"title\":\"Bipartite Ramsey number pairs that involve combinations of cycles and odd paths\",\"authors\":\"Ernst J. 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}","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
摘要
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)。
Bipartite Ramsey number pairs that involve combinations of cycles and odd paths
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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