{"title":"线性森林中的拉姆齐链","authors":"Gary Chartrand, Ritabrato Chatterjee, Ping Zhang","doi":"10.3390/axioms12111019","DOIUrl":null,"url":null,"abstract":"Every red–blue coloring of the edges of a graph G results in a sequence G1, G2, …, Gℓ of pairwise edge-disjoint monochromatic subgraphs Gi (1≤i≤ℓ) of size i, such that Gi is isomorphic to a subgraph of Gi+1 for 1≤i≤ℓ−1. Such a sequence is called a Ramsey chain in G, and ARc(G) is the maximum length of a Ramsey chain in G, with respect to a red–blue coloring c. The Ramsey index AR(G) of G is the minimum value of ARc(G) among all the red–blue colorings c of G. If G has size m, then k+12≤m<k+22 for some positive integer k. It has been shown that there are infinite classes S of graphs, such that for every graph G of size m in S, AR(G)=k if and only if k+12≤m<k+22. Two of these classes are the matchings mK2 and paths Pm+1 of size m. These are both subclasses of linear forests (a forest of which each of the components is a path). It is shown that if F is any linear forest of size m with k+12<m<k+22, then AR(F)=k. Furthermore, if F is a linear forest of size k+12, where k≥4, that has at most k−12 components, then AR(F)=k, while for each integer t with k−12<t<k+12 there is a linear forest F of size k+12 with t components, such that AR(F)=k−1.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"1187 1","pages":"0"},"PeriodicalIF":1.9000,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Ramsey Chains in Linear Forests\",\"authors\":\"Gary Chartrand, Ritabrato Chatterjee, Ping Zhang\",\"doi\":\"10.3390/axioms12111019\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Every red–blue coloring of the edges of a graph G results in a sequence G1, G2, …, Gℓ of pairwise edge-disjoint monochromatic subgraphs Gi (1≤i≤ℓ) of size i, such that Gi is isomorphic to a subgraph of Gi+1 for 1≤i≤ℓ−1. Such a sequence is called a Ramsey chain in G, and ARc(G) is the maximum length of a Ramsey chain in G, with respect to a red–blue coloring c. The Ramsey index AR(G) of G is the minimum value of ARc(G) among all the red–blue colorings c of G. If G has size m, then k+12≤m<k+22 for some positive integer k. It has been shown that there are infinite classes S of graphs, such that for every graph G of size m in S, AR(G)=k if and only if k+12≤m<k+22. Two of these classes are the matchings mK2 and paths Pm+1 of size m. These are both subclasses of linear forests (a forest of which each of the components is a path). It is shown that if F is any linear forest of size m with k+12<m<k+22, then AR(F)=k. Furthermore, if F is a linear forest of size k+12, where k≥4, that has at most k−12 components, then AR(F)=k, while for each integer t with k−12<t<k+12 there is a linear forest F of size k+12 with t components, such that AR(F)=k−1.\",\"PeriodicalId\":53148,\"journal\":{\"name\":\"Axioms\",\"volume\":\"1187 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.9000,\"publicationDate\":\"2023-10-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Axioms\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3390/axioms12111019\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Axioms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/axioms12111019","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
对图G的每条边进行红蓝着色，得到大小为i的对边不相交的单色子图Gi(1≤i≤r)的序列G1, G2，…，G，使得Gi在1≤i≤r−1时与Gi+1的子图同构。这样的序列称为拉姆齐链G,和弧(G)的最大长度是拉姆齐连锁G,对红蓝着色c。拉姆齐指数基于“增大化现实”技术(G)的最小值是G的弧(G)在所有G .如果G大小的红蓝色素c m, k + 12≤m<一些正整数k + 22 k。它已经表明,有无限类的图表,这样每一个图G的大小m S,基于“增大化现实”技术(G) = k当且仅当k + 12≤m< k + 22。其中两个类是匹配大小为m的mK2和路径Pm+1。它们都是线性森林的子类(其中每个组件都是路径)。证明了如果F是任意大小为m且k+12<m<k+22的线性森林，则AR(F)=k。更进一步，如果F是一个大小为k+12的线性森林，其中k≥4，最多有k−12个分量，则AR(F)=k，而对于k−12<t<k+12的每一个整数t，存在一个大小为k+12，有t个分量的线性森林F，使得AR(F)=k−1。
Every red–blue coloring of the edges of a graph G results in a sequence G1, G2, …, Gℓ of pairwise edge-disjoint monochromatic subgraphs Gi (1≤i≤ℓ) of size i, such that Gi is isomorphic to a subgraph of Gi+1 for 1≤i≤ℓ−1. Such a sequence is called a Ramsey chain in G, and ARc(G) is the maximum length of a Ramsey chain in G, with respect to a red–blue coloring c. The Ramsey index AR(G) of G is the minimum value of ARc(G) among all the red–blue colorings c of G. If G has size m, then k+12≤m
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Axiomatic theories in physics and in mathematics (for example, axiomatic theory of thermodynamics, and also either the axiomatic classical set theory or the axiomatic fuzzy set theory) Axiomatization, axiomatic methods, theorems, mathematical proofs Algebraic structures, field theory, group theory, topology, vector spaces Mathematical analysis Mathematical physics Mathematical logic, and non-classical logics, such as fuzzy logic, modal logic, non-monotonic logic. etc. Classical and fuzzy set theories Number theory Systems theory Classical measures, fuzzy measures, representation theory, and probability theory Graph theory Information theory Entropy Symmetry Differential equations and dynamical systems Relativity and quantum theories Mathematical chemistry Automata theory Mathematical problems of artificial intelligence Complex networks from a mathematical viewpoint Reasoning under uncertainty Interdisciplinary applications of mathematical theory.
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