Mark Budden
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View article: Ramsey theory for a generalized fan versus triangles
Ramsey theory for a generalized fan versus triangles Open
In this paper, we consider Ramsey and Gallai-Ramsey numbers for a generalized fan \(F_{t,n}:=K_1+nK_t\) versus triangles. Besides providing some general lower bounds, our main results include the evaluations of \(r(F_{3,2}, K_3)=13\) and \…
View article: Rainbow numbers in planar host graphs
Rainbow numbers in planar host graphs Open
If F is a nonempty set of graphs that contain H as a subgraph, then the rainbow number rb(F,H) is the least t ? N such that every t-coloring of F ? F that uses all t colors contains a rainbow subgraph isomorphic to H. In this paper, we con…
View article: Lower Bounds for Multicolor Star-Critical Ramsey Numbers
Lower Bounds for Multicolor Star-Critical Ramsey Numbers Open
The star-critical Ramsey number is a refinement of the concept of a Ramsey number. In this paper, we give equivalent criteria for which the star-critical Ramsey number vanishes. Next, we provide a new general lower bound for multicolor sta…
View article: Rainbow Numbers for the Generalized Schur Equation $x_1 + x_2 + \cdots + x_{m-1} = x_m$
Rainbow Numbers for the Generalized Schur Equation $x_1 + x_2 + \cdots + x_{m-1} = x_m$ Open
We consider the rainbow Schur number $RS_m(n)$, defined to be the minimum number of colors such that every coloring of $\{1,2,\ldots,n\}$, using all $RS_m(n)$ colors, contains a rainbow solution to the equation $x_1+x_2+\cdots +x_{m-1}=x_m…
View article: The multicolor star-critical Gallai-Ramsey number for a path of order 5
The multicolor star-critical Gallai-Ramsey number for a path of order 5 Open
In this paper, the t-color star-critical Gallai-Ramsey number for a path of order 5 is determined.It is proved that t + 1 edges are both necessary and sufficient to add between a vertex and a critical coloring for the t-color Gallai-Ramsey…
View article: Ramsey Numbers for Connected 2-Colorings of Complete Graphs
Ramsey Numbers for Connected 2-Colorings of Complete Graphs Open
In 1978, David Sumner introduced a variation of Ramsey numbers by restricting to 2-colorings in which the subgraphs spanned by edges in each color are connected. This paper continues the study of connected Ramsey numbers, including the eva…
View article: Star-critical Gallai-Ramsey numbers involving the disjoint union of triangles
Star-critical Gallai-Ramsey numbers involving the disjoint union of triangles Open
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 ar…
View article: Multicolor star-critical Ramsey numbers and Ramsey-good graphs
Multicolor star-critical Ramsey numbers and Ramsey-good graphs Open
This paper seeks to develop the multicolor version of star-critical Ramsey numbers, which serve as a measure of the strength of the corresponding Ramsey numbers. We offer several general theorems, some of which focus on Ramsey-good cases (…
View article: Anti-Ramsey Hypergraph Numbers
Anti-Ramsey Hypergraph Numbers Open
The anti-Ramsey number arn(H) of an r-uniform hypergraph is the maximum number of colors that can be used to color the hyperedges of a complete r-uniform hypergraph on n vertices without producing a rainbow copy of H. In this paper, we det…
View article: Sierpiński products of r-uniform hypergraphs
Sierpiński products of r-uniform hypergraphs Open
If H1 and H2 are r-uniform hypergraphs and f is a function from the set of all (r − 1)-element subsets of V(H1) into V(H2), then the Sierpiński product H1⊗fH2 is defined to have vertex set V(H1) × V(H2) and hyperedges falling into two clas…
View article: Algebraic Properties of a Hypergraph Lifting Map
Algebraic Properties of a Hypergraph Lifting Map Open
Recent work in hypergraph Ramsey theory has involved the introduction of a "lifting map" that associates a certain $3$-uniform hypergraph to a given graph, bounding cliques in a predictable way. In this paper, we interpret the lifting map …
View article: Schur numbers involving rainbow colorings
Schur numbers involving rainbow colorings Open
In this paper, we introduce two different generalizations of Schur numbers that involve rainbow colorings. Motivated by well-known generalizations of Ramsey numbers, we first define the rainbow Schur number RS(n) to be the minimum number o…
View article: Constructive Methods in Gallai-Ramsey Theory for Hypergraphs
Constructive Methods in Gallai-Ramsey Theory for Hypergraphs Open
Much recent progress in hypergraph Ramsey theory has focused on constructions that lead to lower bounds for the corresponding Ramsey numbers. In this paper, we consider applications of these results to Gallai colorings. That is, we focus o…
View article: Minimally Connected Hypergraphs
Minimally Connected Hypergraphs Open
Graphs and hypergraphs are foundational structures in discrete mathematics. They have many practical applications, including the rapidly developing field of bioinformatics, and more generally, biomathematics. They are also a source of inte…
View article: Weakened Gallai-Ramsey numbers
Weakened Gallai-Ramsey numbers Open
In the Ramsey theory of graphs, one seeks to determine the value of the Ramsey number rt(n), defined to be the least natural number p such that every coloring of the edges of Kp using t colors results in a monochromatic copy of Kn in some …
View article: Trees and $n$-Good Hypergraphs
Trees and $n$-Good Hypergraphs Open
Trees fill many extremal roles in graph theory, being minimally connected and serving a critical role in the definition of $n$-good graphs. In this article, we consider the generalization of trees to the setting of $r$-uniform hypergraphs …
View article: Weakened Ramsey Numbers and Their Hypergraph Analogues
Weakened Ramsey Numbers and Their Hypergraph Analogues Open
See the abstract in the attached pdf.
View article: The lifting of graphs to 3-uniform hypergraphs and some applications to hypergraph Ramsey theory
The lifting of graphs to 3-uniform hypergraphs and some applications to hypergraph Ramsey theory Open
Given a simple graph [math] , we describe a “lifting” to a [math] -uniform hypergraph [math] that sends the complement of [math] to the complement of [math] . We consider the effects of this lifting on cycles, complete subhypergraphs, and …