The Goodness of Long Path With Respect to Multiple Copies of Complete Graphs

Sudarsana, I Wayan

doi:10.22342/jims.20.1.179.31-35

Cited by 4 publications

(4 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Certainly, some results provide evidence for that the answer of this question is yes for some F . Sudarsana [14] showed that P n with n ≥ (t − 2)((tm − 2)(m − 1) + 1) + 3 is tK m -good for m and t ≥ 2 be integers. Indeed, the condition on the number of vertices n can not be released completely.…”

Section: Remarksmentioning

confidence: 99%

The Ramsey Number for a Forest versus Disjoint Union of Complete Graphs

Hu¹,

Peng²

2022

Preprint

View full text Add to dashboard Cite

Given two graphs G and H, the Ramsey number R(G, H) is the minimum integer N such that any coloring of the edges of KN in red or blue yields a red G or a blue H. Let v(G) be the number of vertices of G and χ(G) be the chromatic number of G. Let s(G) denote the chromatic surplus of G, the cardinality of a minimum color class taken over all proper colorings of G with χ(G) colors. Burr [3] showed that for a connected graph G and a graph Chvátal [5] showed that any tree is Km-good for m ≥ 2, where Km denotes a complete graph with m vertices. By applying this result, Stahl [13] determined the Ramsey number of a forest versus Km. Concerning whether a tree is H-good for H being disjoint union of complete graphs, Chvátal and Harary [6] showed that any tree is 2K2-good, where tH denotes the union of t disjoint copies of graph H. Sudarsana, Adiwijaya and Musdalifah [15] proved that the n-vertex path Pn is 2Km-good for n ≥ 3 and m ≥ 2, and conjectured that any tree Tn with n vertices is 2Km-good. Recently, Pokrovskiy and Sudakov [11] proved that Pn is H-good for a graph with n ≥ 4v(H). Balla, Pokrovskiy and Sudakov [2] showed that for all ∆ and k, there exists a constant C ∆,k such that for any tree T with maximum degree at most ∆ and any H withIn this paper, we explore the Ramsey number of forest versus disjoint union of complete graphs. We first confirm the conjecture by Sudarsana, Adiwijaya and Musdalifah [15] that any tree is 2Kmgood for n ≥ 3 and m ≥ 2. A key proposition in our proof is that a tree with n vertices can be obtained from any tree with n vertices by performing a series of "Stretching" and "Expanding" operations. On this foundation, we show that these two operations preserve the "2Km-goodness" property, and confirm that any tree is 2Km-good. We also prove a conclusion which yields that Tn is Km ∪ K l -good, where Km ∪ K l is the disjoint union of Km and K l , m > l ≥ 2. Furthermore, we extend the Ramsey goodness of connected graphs to disconnected graphs and study the relation between the Ramsey number of the components of a disconnected graph F versus a graph H. We show that if each component of a graph F is H-good, then F is H-good. Our result implies the exact value of R(F, Km ∪ K l ), where F is a forest and m, l ≥ 2. It's interesting to explore what kind of graph F can satisfy that F is H-good and G-good implies that F is H ∪ G-good when the number of vertices in F is sufficiently large.

show abstract

Section: Remarksmentioning

confidence: 99%

The Ramsey Number for a Forest versus Disjoint Union of Complete Graphs

Hu¹,

Peng²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…By Theorems 3, 2 and 4, we have the assertion. [17]) If m, k ≥ 2 and n ≥ (k −2)((km −2)(m −1)+1)+3, then K a 1 ,a 2 ,...,a m . By Theorems 3, 2 and 4, we have the assertion.…”

Section: Theorem 9 (Pokrovskiymentioning

confidence: 99%

On the Ramsey-Goodness of Paths

Bielak

2016

Graphs and Combinatorics

View full text Add to dashboard Cite

“…The disjoint union of graphs, t i=1 G i , is a graph with the vertex set t i=1 V i and the edge set t i=1 E i ; and if G i G for each i then t i=1 G i = tG. For the definition of Ramsey number, we cite the results proposed by Sudarsana [13], [11], [12]; that is, for graphs G and H, the Ramsey number R(G, H) is the smallest natural number n such that in every red and blue colorings of the edges of the complete graph K n , there is a graph G in K n which is all edges are red or a graph H in K n which is all edges are blue. In the other terminology, a graph F is called (G, H)-free, if F contains no G and F contains no H. Furthermore, we have an equivalent definition of Ramsey number R(G, H), that is, the smallest positive integer n such that there is no (G, H)-free graph on n vertices exists.…”

Section: Introductionmentioning

confidence: 99%

“…The notation tW m discribe a graph with t copies of wheels of order m + 1. Surahmat et al [14] proved that cycle C n is W m -good; Chen et al [6] showed that P n is W m -good for even m and n ≥ m − 1 ≥ 3; P n is tW 4 -good for n ≥ 15t 2 − 4t + 2, t ≥ 1 [13], and Sudarsana [12] recently proved that C n is tK m -good. Meanwhile, S n is not W 6 -good for n ≥ 3 [5].…”

Section: Introductionmentioning

confidence: 99%

A note on the Ramsey number for cycle with respect to multiple copies of wheels

Sudarsana

2021

EJGTA

Self Cite

View full text Add to dashboard Cite

Let K n be a complete graph with n vertices. For graphs G and H, the Ramsey number R(G, H) is the smallest positive integer n such that in every red-blue coloring on the edges of K n , there is a red copy of graph G or a blue copy of graph H in K n . Determining the Ramsey number R(C n , tW m ) for any integers t ≥ 1, n ≥ 3 and m ≥ 4 in general is a challenging problem, but we conjecture that for any integers t ≥ 1 and m ≥ 4, there exists n 0 = f (t, m) such that cycle C n is tW m -good for any n ≥ n 0 . In this paper, we provide some evidence for the conjecture in the case of m = 4 that if n ≥ n 0 then the Ramsey number R(C n , tW 4 ) = 2n + t − 2 with n 0 = 15t 2 − 4t + 2 and t ≥ 1. Furthermore, if G is a disjoint union of cycles then the Ramsey number R(G, tW 4 ) is also derived.

show abstract

The Goodness of Long Path With Respect to Multiple Copies of Complete Graphs

Cited by 4 publications

References 7 publications

The Ramsey Number for a Forest versus Disjoint Union of Complete Graphs

The Ramsey Number for a Forest versus Disjoint Union of Complete Graphs

On the Ramsey-Goodness of Paths

A note on the Ramsey number for cycle with respect to multiple copies of wheels

Contact Info

Product

Resources

About