Download or read book Hadwiger Numbers and Gallai Ramsey Numbers of Special Graphs written by Christian Bosse and published by . This book was released on 2019 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation explores two separate topics on graphs.
Download or read book Topics in Gallai Ramsey Theory written by Colton Magnant and published by Springer Nature. This book was released on 2020-07-04 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores topics in Gallai-Ramsey theory, which looks into whether rainbow colored subgraphs or monochromatic subgraphs exist in a sufficiently large edge-colored complete graphs. A comprehensive survey of all known results with complete references is provided for common proof methods. Fundamental definitions and preliminary results with illustrations guide readers to comprehend recent innovations. Complete proofs and influential results are discussed with numerous open problems and conjectures. Researchers and students with an interest in edge-coloring, Ramsey Theory, and colored subgraphs will find this book a valuable guide for entering Gallai-Ramsey Theory.
Download or read book Gallai Ramsey Numbers for Graphs and Their Generalizations written by Xihe Li and published by . This book was released on 2021 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Star Critical Ramsey Numbers for Graphs written by Mark R. Budden and published by Springer Nature. This book was released on 2023-05-13 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is a comprehensive survey of the literature surrounding star-critical Ramsey numbers. First defined by Jonelle Hook in her 2010 dissertation, these numbers aim to measure the sharpness of the corresponding Ramsey numbers by determining the minimum number of edges needed to be added to a critical graph for the Ramsey property to hold. Despite being in its infancy, the topic has gained significant attention among Ramsey theorists. This work provides researchers and students with a resource for studying known results and their complete proofs. It covers typical results, including multicolor star-critical Ramsey numbers for complete graphs, trees, cycles, wheels, and n-good graphs, among others. The proofs are streamlined and, in some cases, simplified, with a few new results included. The book also explores the connection between star-critical Ramsey numbers and deleted edge numbers, which focus on destroying the Ramsey property by removing edges. The book concludes with open problems and conjectures for researchers to consider, making it a valuable resource for those studying the field of star-critical Ramsey numbers.
Download or read book On Gallai ramsey Numbers of Complete Bipartite Graphs written by Seth Gutierrez and published by . This book was released on 2017 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt: Author's abstract: We show the minimum number of vertices necessary of a complete Gallai-colored graph on $k$ colors that forces a monochromatic complete bipartite graph $K_{2,m}$, $K_{3,3}$, and $K_{3,m}$, for $m\geq 3$. We also provide some results on graphs that are free of rainbow triangles with a pendant edge that provide more insight to the structure of Gallai colorings.
Download or read book Ramsey Theory written by Alexander Soifer and published by Springer Science & Business Media. This book was released on 2010-10-29 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the theory’s history, recent developments, and some promising future directions through invited surveys written by prominent researchers in the field. The first three surveys provide historical background on the subject; the last three address Euclidean Ramsey theory and related coloring problems. In addition, open problems posed throughout the volume and in the concluding open problem chapter will appeal to graduate students and mathematicians alike.
Download or read book The Triangle Free Process and the Ramsey Number R 3 k written by Gonzalo Fiz Pontiveros and published by American Mathematical Soc.. This book was released on 2020-04-03 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: The areas of Ramsey theory and random graphs have been closely linked ever since Erdős's famous proof in 1947 that the “diagonal” Ramsey numbers R(k) grow exponentially in k. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the “off-diagonal” Ramsey numbers R(3,k). In this model, edges of Kn are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted Gn,△. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kim's celebrated result that R(3,k)=Θ(k2/logk). In this paper the authors improve the results of both Bohman and Kim and follow the triangle-free process all the way to its asymptotic end.
Download or read book Gallai Ramsey Number of an 8 cycle written by Jonathan Gregory and published by . This book was released on 2016 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt: Author's abstract: Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge-coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we establish the Gallai-Ramsey number of an 8-cycle for all positive integers.
Download or read book Generalised Ramsey numbers and Bruhat order on involutions written by Mikael Hansson and published by Linköping University Electronic Press. This book was released on 2015-12-03 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis consists of two papers within two different areas of combinatorics. Ramsey theory is a classic topic in graph theory, and Paper A deals with two of its most fundamental problems: to compute Ramsey numbers and to characterise critical graphs. More precisely, we study generalised Ramsey numbers for two sets ?1 and ?2 of cycles. We determine, in particular, all generalised Ramsey numbers R(?1, ?2) such that ?1 or ?2 contains a cycle of length at most 6, or the shortest cycle in each set is even. This generalises previous results of Erdös, Faudree, Rosta, Rousseau, and Schelp. Furthermore, we give a conjecture for the general case. We also characterise many (?1, ?2)-critical graphs. As special cases, we obtain complete characterisations of all (Cn,C3)-critical graphs for n ? 5, and all (Cn,C5)-critical graphs for n ? 6. In Paper B, we study the combinatorics of certain partially ordered sets. These posets are unions of conjugacy classes of involutions in the symmetric group Sn, with the order induced by the Bruhat order on Sn. We obtain a complete characterisation of the posets that are graded. In particular, we prove that the set of involutions with exactly one fixed point is graded, which settles a conjecture of Hultman in the affirmative. When the posets are graded, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, recently proved by Can, Cherniavsky, and Twelbeck.
Download or read book Graph Related Ramsey Numbers written by Ruth Anne Hendrix and published by . This book was released on 1990 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Generalization of Ramsey Theory for Graphs written by Chung Laung Liu and published by . This book was released on 1977 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Ramsey Numbers for Sets of Five Vertex Graphs with Fixed Numbers of Edges written by Heiko Harborth and published by . This book was released on 1992 with total page 14 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Multicolor Ramsey and List Ramsey Numbers for Double Stars written by Jake Ruotolo and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. For graphs G and H we write G [rightwards arrow] (H; k) if every k-edge-coloring of G contains a monochromatic copy of H in color i for some i [element of] {1,2,...,k}. Similarly, we write G [rightwards arrow with stroke] (H; k) if there exists a k-edge-coloring of G with no monochromatic H. Such a coloring c is a critical k-coloring. The k-color Ramsey number of the graph H, denoted r(H; k), is the smallest integer N such that K[subscript N][rightwards arrow] (H; k), where K[subscript N] is the complete graph on N vertices. Despite active research for decades, very little is known about Ramsey numbers of graphs. This is especially true for r(H; k) when k [greater than or equal to] 3, also known as the multicolor Ramsey number of H. Let S[subscript n] denote the star on n + 1 vertices, the graph with one vertex of degree n (the center of S[subscript n]), and n vertices of degree 1. The double star S(n,m), where n [greater than or equal to] m [greater than or equal to] 1, is the graph consisting of the disjoint union of two stars S[subscript n] and S[subscript m] together with an edge joining their centers. In this thesis, we study the multicolor Ramsey number of double stars. We obtain upper and lower bounds for r(S(n,m); k) when k [greater than or equal to] 3 and prove that r(S(n,m); k) = nk + m + 2 when k [greater than or equal to] 3 is odd and n is sufficiently large. We also investigate a generalization of the Ramsey number known as the list Ramsey number. Let L : E(K[subscript n]) [rightwards arrow] ([Natural number over k]) be an assignment of k-element subsets of [Natural number] to the edges of K[subscript n]. A coloring c : E(K[subscript n]) [rightwards arrow] [Natural number] is said to be an L-coloring if c(e) [element of] L(e) for all e [element of] E(K[subscript n]). The k-color list Ramsey number r[mathematical subscript ell] (H; k) of a graph H is defined as the smallest n such that there is some L : E(K[subscript n]) [rightwards arrow] [Natural number over k] for which every L-coloring of K[subscript n] contains a monochromatic copy of H. In this thesis, we study r[mathematical subscript ell](S(1, 1); p) and r[mathematical subscript ell](S[subscript n]; p) where p is an odd prime number.
Download or read book Group action Graphs and Ramsey Graph Theory written by Scott Paul Stevens and published by . This book was released on 1987 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Irredundant Ramsey Numbers for Graphs written by Richard Charles Brewster and published by . This book was released on 1988 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Survey of Generalized Ramsey Numbers for Graphs written by Bryce D. Morgan and published by . This book was released on 1975 with total page 74 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Two New Ramsey Numbers written by James N. McNamara and published by . This book was released on 1992 with total page 74 pages. Available in PDF, EPUB and Kindle. Book excerpt: "A graph with many vertices cannot be homogeneous, i .e., for any pair of integers (i, j) all large graphs must contain either a complete subgraph on i vertices or an independent set of size j. The Ramsey number for (i, j) is the smallest integer R such that all graphs with at least R vertices have this property. For example, the (3,3) Ramsey number is 6; if a graph has 6 or more vertices, then is must contain a triangle or an independent set of size 3. The (4,4) Ramsey number is 18, found in 1954 [GG] . The (5,5) Ramsey number is still unknown; it is between 43 and 52. This thesis deals with subgraphs slightly different from the classical types. The subgraphs here are complete graphs with one edge missing and induced subgraphs with exactly one edge. The (4,6) and (4,7) Ramsey numbers for these types of subgraphs is computed. The method used is an exhaustive search, with many shortcuts employed to reduce computation time."--Abstract.
