Critical graphs for R(P_n,P_m) and the star-critical Ramsey number for paths

Hook, Jonelle

doi:10.7151/dmgt.1827

Cited by 20 publications

(6 citation statements)

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“…First introduced by Hook [9] in 2010, and developed in [10] and [11], the star-critical Ramsey number of a collection of graphs focuses on the number of edges that must be added between a vertex and a critical coloring of a certain complete graph to guarantee the Ramsey property. In a sense, it measures the strength of the corresponding Ramsey number.…”

Section: Introductionmentioning

confidence: 99%

Star-critical Gallai-Ramsey numbers involving the disjoint union of triangles

Budden

Daouda

2022

Art Discrete Appl. Math.

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In the Ramsey theory of graphs, star-critical Gallai-Ramsey numbers serve as a measure of the strength of their corresponding Gallai-Ramsey numbers. In this paper, we evaluate some classes of star-critical Gallai-Ramsey numbers when the arguments are the disjoint unions of K 3 -subgraphs. The methods used are constructive and in all cases considered, we find that the removal of a single edge destroys the Ramsey property.

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Section: Introductionmentioning

confidence: 99%

Star-critical Gallai-Ramsey numbers involving the disjoint union of triangles

Budden

Daouda

2022

Art Discrete Appl. Math.

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“…Over the years, many generalisations of Ramsey numbers have been considered (the survey [11] contains many examples); in particular, determining the generalised Ramsey number of a path in a variety of these more general settings has often turned out to be a very interesting problem [1, 4–7, 12, 21, 22, 30]. A particularly natural generalisation first considered by Erdős, Hajnal and Rado in 1965 [13] asks the following.…”

Section: Introductionmentioning

confidence: 99%

The (t−1) $(t-1)$‐chromatic Ramsey number for paths

Bucić

Khamseh

2022

Journal of Graph Theory

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The following relaxation of the classical problem of determining the Ramsey number of a fixed graph has first been proposed by Erdős, Hajnal and Rado over 50 years ago. Given a graph G $G$ and an integer t ≥ 2 $t\ge 2$ determine the minimum number N $N$ such that in any t $t$‐coloured complete graph on N $N$ vertices there is a copy of G $G$ using only edges of some t − 1 $t-1$ colours. We determine the answer precisely when G $G$ is a path.

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“…In other words, this is the largest star that can be removed from K r yet every 2-coloring is still forced to contain either a red G 1 or a blue G 2 . The paper [8] studies critical graphs for r(P i 1 , P i 2 ) and the starcritical ramsey number for paths. In [18] the authors find the maximum star deleted from Ramsey graphs of B m = K 2 + mK 1 and T n , where T n is a tree of order n. On the other hand, the complete bipartite-critical Ramsey number introduced in [15] as…”

Section: Introductionmentioning

confidence: 99%

A note on (t - 1)-chromatic Ramsey number of linear forests

Khamseh

2020

International Journal of Computer Mathematics: Computer Systems

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Given t distinct colors, we order the t subsets of t − 1 colors in some arbitrary manner. Let G 1 , G 2 , . . . , G t be graphs. The (t−1)-chromatic Ramsey number, denoted by r t t−1 (G 1 , G 2 , . . . , G t ), is defined to be the least number n such that if the edges of the complete graph K n are colored in any fashion with t colors, then for some i the subgraph whose edges are colored with the ith subset of colors contains a copy of G i . In this paper, we find the value of r t t−1 (G 1 , G 2 , . . . , G t ) when each G i is a path. 1

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Critical graphs for R(P_n,P_m) and the star-critical Ramsey number for paths

Cited by 20 publications

References 1 publication

Star-critical Gallai-Ramsey numbers involving the disjoint union of triangles

Star-critical Gallai-Ramsey numbers involving the disjoint union of triangles

The (t−1) $(t-1)$‐chromatic Ramsey number for paths

A note on (t - 1)-chromatic Ramsey number of linear forests

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