Critical Graphs for R(P n , P m ) and the Star-Critical Ramsey Number for Paths

Jonelle Hook

Discussiones Mathematicae Graph Theory (2015)

  • Volume: 35, Issue: 4, page 689-701
  • ISSN: 2083-5892

Abstract

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The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H. The star-critical Ramsey number r∗(G,H) is the smallest integer k such that every 2-coloring of the edges of Kr − K1,r−1−k contains either a red copy of G or a blue copy of H. We will classify the critical graphs, 2-colorings of the complete graph on R(G,H) − 1 vertices with no red G or blue H, for the path-path Ramsey number. This classification will be used in the proof of r∗(Pn, Pm).

How to cite

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Jonelle Hook. " Critical Graphs for R(P n , P m ) and the Star-Critical Ramsey Number for Paths ." Discussiones Mathematicae Graph Theory 35.4 (2015): 689-701. <http://eudml.org/doc/275867>.

@article{JonelleHook2015,
abstract = {The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H. The star-critical Ramsey number r∗(G,H) is the smallest integer k such that every 2-coloring of the edges of Kr − K1,r−1−k contains either a red copy of G or a blue copy of H. We will classify the critical graphs, 2-colorings of the complete graph on R(G,H) − 1 vertices with no red G or blue H, for the path-path Ramsey number. This classification will be used in the proof of r∗(Pn, Pm).},
author = {Jonelle Hook},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {Ramsey number; critical graph; star-critical Ramsey number; path},
language = {eng},
number = {4},
pages = {689-701},
title = { Critical Graphs for R(P n , P m ) and the Star-Critical Ramsey Number for Paths },
url = {http://eudml.org/doc/275867},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Jonelle Hook
TI - Critical Graphs for R(P n , P m ) and the Star-Critical Ramsey Number for Paths
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 4
SP - 689
EP - 701
AB - The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H. The star-critical Ramsey number r∗(G,H) is the smallest integer k such that every 2-coloring of the edges of Kr − K1,r−1−k contains either a red copy of G or a blue copy of H. We will classify the critical graphs, 2-colorings of the complete graph on R(G,H) − 1 vertices with no red G or blue H, for the path-path Ramsey number. This classification will be used in the proof of r∗(Pn, Pm).
LA - eng
KW - Ramsey number; critical graph; star-critical Ramsey number; path
UR - http://eudml.org/doc/275867
ER -

References

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  1. [1] L. Gerencsér and A. Gyárfás, On Ramsey-type problems, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 10 (1967) 167-170. Zbl0163.45502
  2. [2] J. Hook, The classification of critical graphs and star-critical Ramsey numbers (Ph.D. Thesis, Lehigh University, 2010). 
  3. [3] J. Hook and G. Isaak, Star-critical Ramsey numbers, Discrete Appl. Math. 159 (2011) 328-334. doi:10.1016/j.dam.2010.11.007[Crossref] Zbl1207.05212
  4. [4] R.J. Faudree, S.L. Lawrence, T.D. Parsons and R.H. Schelp, Path-cycle Ramsey numbers, Discrete Math. 10 (1974) 269-277. doi:10.1016/0012-365X(74)90122-8 Zbl0293.05120
  5. [5] R.J. Faudree and R.H. Schelp, All Ramsey numbers for cycles in graphs, Discrete Math. 8 (1974) 313-329. doi:10.1016/0012-365X(74)90151-4 Zbl0294.05122
  6. [6] D.B. West, Introduction to Graph Theory (Second Ed., Prentice Hall, Upper Saddle River, New Jersey, 2000). 

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