The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths

Halina Bielak; Kinga Dąbrowska

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2015)

  • Volume: 69, Issue: 2
  • ISSN: 0365-1029

Abstract

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The Ramsey number R ( G , H ) for a pair of graphs G and H is defined as the smallest integer n such that, for any graph F on n vertices, either F contains G or F ¯ contains H as a subgraph, where F ¯ denotes the complement of F . We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers R ( K 1 + L n , P m ) and R ( K 1 + L n , C m ) for some integers m , n , where L n is a linear forest of order n with at least one edge.

How to cite

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Halina Bielak, and Kinga Dąbrowska. "The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 69.2 (2015): null. <http://eudml.org/doc/289826>.

@article{HalinaBielak2015,
abstract = {The Ramsey number $R(G, H)$ for a pair of graphs $G$ and $H$ is defined as the smallest integer $n$ such that, for any graph $F$ on $n$ vertices, either $F$ contains $G$ or $\overline\{F\}$ contains $H$ as a subgraph, where $\overline\{F\}$ denotes the complement of $F$. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers $R(K_1+L_n, P_m)$ and $R(K_1+L_n, C_m)$ for some integers $m$, $n$, where $L_n$ is a linear forest of order $n$ with at least one edge.},
author = {Halina Bielak, Kinga Dąbrowska},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Cycle; path; Ramsey number; Turan number},
language = {eng},
number = {2},
pages = {null},
title = {The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths},
url = {http://eudml.org/doc/289826},
volume = {69},
year = {2015},
}

TY - JOUR
AU - Halina Bielak
AU - Kinga Dąbrowska
TI - The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2015
VL - 69
IS - 2
SP - null
AB - The Ramsey number $R(G, H)$ for a pair of graphs $G$ and $H$ is defined as the smallest integer $n$ such that, for any graph $F$ on $n$ vertices, either $F$ contains $G$ or $\overline{F}$ contains $H$ as a subgraph, where $\overline{F}$ denotes the complement of $F$. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers $R(K_1+L_n, P_m)$ and $R(K_1+L_n, C_m)$ for some integers $m$, $n$, where $L_n$ is a linear forest of order $n$ with at least one edge.
LA - eng
KW - Cycle; path; Ramsey number; Turan number
UR - http://eudml.org/doc/289826
ER -

References

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  17. Surahmat, Baskoro, E. T., Broersma, H. J., The Ramsey numbers of large cycles versus small wheels, Integers 4 (2004), A10. 
  18. Surahmat, Baskoro, E. T., Tomescu, I., The Ramsey numbers of large cycles versus odd wheels, Graphs Combin. 24 (2008), 53-58. 
  19. Surahmat, Baskoro, E. T., Tomescu, I., The Ramsey numbers of large cycles versus wheels, Discrete Math. 306 (24) (2006), 3334-3337. 
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