Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths

Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths

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...

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Main Author: Hook Jonelle
Format: Article
Language:English
Published: Sciendo 2015-11-01
Series:Discussiones Mathematicae Graph Theory
Subjects:
ramsey number
critical graph
star-critical ramsey number
path
Online Access:https://doi.org/10.7151/dmgt.1827
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spelling doaj-522f1d2b4f5741c8903d36909a19d3a92021-09-05T17:20:21ZengSciendoDiscussiones Mathematicae Graph Theory2083-58922015-11-0135468970110.7151/dmgt.1827dmgt.1827Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for PathsHook Jonelle0Mount St. Mary’s University Department of Mathematics and Computer Science Emmitsburg, MD 21727The 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).https://doi.org/10.7151/dmgt.1827ramsey numbercritical graphstar-critical ramsey numberpath
collection DOAJ
language English
format Article
sources DOAJ
author Hook Jonelle
spellingShingle Hook Jonelle
Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths
Discussiones Mathematicae Graph Theory
ramsey number
critical graph
star-critical ramsey number
path
author_facet Hook Jonelle
author_sort Hook Jonelle
title Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths
title_short Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths
title_full Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths
title_fullStr Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths
title_full_unstemmed Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths
title_sort critical graphs for r(pn, pm) and the star-critical ramsey number for paths
publisher Sciendo
series Discussiones Mathematicae Graph Theory
issn 2083-5892
publishDate 2015-11-01
description 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).
topic ramsey number
critical graph
star-critical ramsey number
path
url https://doi.org/10.7151/dmgt.1827
work_keys_str_mv AT hookjonelle criticalgraphsforrpnpmandthestarcriticalramseynumberforpaths
_version_ 1717786505461628928

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