A Note on On-Line Ramsey Numbers of Stars and Paths
An on-line Ramsey game is a game between two players, Builder and Painter, on an initially empty graph with unbounded set of vertices. In each round, Builder selects two vertices and joins them with an edge while Painter colours the edge immediately with either red or blue. Builder’s aim is to force...
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| Veröffentlicht in: | Bulletin of the Malaysian Mathematical Sciences Society 2021, Vol.44 (5), p.3511-3521 |
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| container_title | Bulletin of the Malaysian Mathematical Sciences Society |
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| creator | Mohd Latip, Fatin Nur Nadia Binti Tan, Ta Sheng |
| description | An on-line Ramsey game is a game between two players, Builder and Painter, on an initially empty graph with unbounded set of vertices. In each round, Builder selects two vertices and joins them with an edge while Painter colours the edge immediately with either red or blue. Builder’s aim is to force either a red copy of a fixed graph
G
or a blue copy of a fixed graph
H
. The game ends with Builder’s victory when Builder manages to force either a red
G
or a blue
H
. The minimum number of rounds for Builder to win the game, regardless of Painter’s strategy, is the on-line Ramsey number
r
~
(
G
,
H
)
. This note focuses on the case when
G
and
H
are stars and paths. In particular, we will prove the upper bound of
r
~
(
S
3
,
P
l
+
1
)
≤
5
l
/
3
+
2
. We will also present an alternative proof for the upper bound of
r
~
(
S
2
=
P
3
,
P
l
+
1
)
=
⌈
5
l
/
4
⌉
. |
| doi_str_mv | 10.1007/s40840-021-01130-x |
| format | Article |
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G
or a blue copy of a fixed graph
H
. The game ends with Builder’s victory when Builder manages to force either a red
G
or a blue
H
. The minimum number of rounds for Builder to win the game, regardless of Painter’s strategy, is the on-line Ramsey number
r
~
(
G
,
H
)
. This note focuses on the case when
G
and
H
are stars and paths. In particular, we will prove the upper bound of
r
~
(
S
3
,
P
l
+
1
)
≤
5
l
/
3
+
2
. We will also present an alternative proof for the upper bound of
r
~
(
S
2
=
P
3
,
P
l
+
1
)
=
⌈
5
l
/
4
⌉
.</description><identifier>ISSN: 0126-6705</identifier><identifier>EISSN: 2180-4206</identifier><identifier>DOI: 10.1007/s40840-021-01130-x</identifier><language>eng</language><publisher>Singapore: Springer Singapore</publisher><subject>Apexes ; Applications of Mathematics ; Games ; Graph coloring ; Graph theory ; Mathematics ; Mathematics and Statistics ; Upper bounds</subject><ispartof>Bulletin of the Malaysian Mathematical Sciences Society, 2021, Vol.44 (5), p.3511-3521</ispartof><rights>Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2021</rights><rights>Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-954f54dde8eceed28ce70c8f47d3106a29b93381e85740da28604ba57a0743ac3</citedby><cites>FETCH-LOGICAL-c319t-954f54dde8eceed28ce70c8f47d3106a29b93381e85740da28604ba57a0743ac3</cites><orcidid>0000-0002-5739-3242</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s40840-021-01130-x$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s40840-021-01130-x$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Mohd Latip, Fatin Nur Nadia Binti</creatorcontrib><creatorcontrib>Tan, Ta Sheng</creatorcontrib><title>A Note on On-Line Ramsey Numbers of Stars and Paths</title><title>Bulletin of the Malaysian Mathematical Sciences Society</title><addtitle>Bull. Malays. Math. Sci. Soc</addtitle><description>An on-line Ramsey game is a game between two players, Builder and Painter, on an initially empty graph with unbounded set of vertices. In each round, Builder selects two vertices and joins them with an edge while Painter colours the edge immediately with either red or blue. Builder’s aim is to force either a red copy of a fixed graph
G
or a blue copy of a fixed graph
H
. The game ends with Builder’s victory when Builder manages to force either a red
G
or a blue
H
. The minimum number of rounds for Builder to win the game, regardless of Painter’s strategy, is the on-line Ramsey number
r
~
(
G
,
H
)
. This note focuses on the case when
G
and
H
are stars and paths. In particular, we will prove the upper bound of
r
~
(
S
3
,
P
l
+
1
)
≤
5
l
/
3
+
2
. We will also present an alternative proof for the upper bound of
r
~
(
S
2
=
P
3
,
P
l
+
1
)
=
⌈
5
l
/
4
⌉
.</description><subject>Apexes</subject><subject>Applications of Mathematics</subject><subject>Games</subject><subject>Graph coloring</subject><subject>Graph theory</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Upper bounds</subject><issn>0126-6705</issn><issn>2180-4206</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LAzEQhoMoWGr_gKeA5-jkO3ssxS8orfhxDtndWW2xuzXZQvvvTV3Bm3OZObzPO_AQcsnhmgPYm6TAKWAgOAPOJbD9CRkJ7oApAeaUjIALw4wFfU4mKa0hjzbCCD4ickoXXY-0a-myZfNVi_Q5bBIe6GK3KTEm2jX0pQ_5CG1Nn0L_kS7IWRM-E05-95i83d2-zh7YfHn_OJvOWSV50bNCq0arukaHFWItXIUWKtcoW0sOJoiiLKR0HJ22CuognAFVBm0DWCVDJcfkaujdxu5rh6n3624X2_zSC62PlOUup8SQqmKXUsTGb-NqE-LBc_BHP37w47Mf_-PH7zMkByjlcPuO8a_6H-obvR5lrg</recordid><startdate>2021</startdate><enddate>2021</enddate><creator>Mohd Latip, Fatin Nur Nadia Binti</creator><creator>Tan, Ta Sheng</creator><general>Springer Singapore</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-5739-3242</orcidid></search><sort><creationdate>2021</creationdate><title>A Note on On-Line Ramsey Numbers of Stars and Paths</title><author>Mohd Latip, Fatin Nur Nadia Binti ; Tan, Ta Sheng</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-954f54dde8eceed28ce70c8f47d3106a29b93381e85740da28604ba57a0743ac3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Apexes</topic><topic>Applications of Mathematics</topic><topic>Games</topic><topic>Graph coloring</topic><topic>Graph theory</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Upper bounds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Mohd Latip, Fatin Nur Nadia Binti</creatorcontrib><creatorcontrib>Tan, Ta Sheng</creatorcontrib><collection>CrossRef</collection><jtitle>Bulletin of the Malaysian Mathematical Sciences Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Mohd Latip, Fatin Nur Nadia Binti</au><au>Tan, Ta Sheng</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Note on On-Line Ramsey Numbers of Stars and Paths</atitle><jtitle>Bulletin of the Malaysian Mathematical Sciences Society</jtitle><stitle>Bull. Malays. Math. Sci. Soc</stitle><date>2021</date><risdate>2021</risdate><volume>44</volume><issue>5</issue><spage>3511</spage><epage>3521</epage><pages>3511-3521</pages><issn>0126-6705</issn><eissn>2180-4206</eissn><abstract>An on-line Ramsey game is a game between two players, Builder and Painter, on an initially empty graph with unbounded set of vertices. In each round, Builder selects two vertices and joins them with an edge while Painter colours the edge immediately with either red or blue. Builder’s aim is to force either a red copy of a fixed graph
G
or a blue copy of a fixed graph
H
. The game ends with Builder’s victory when Builder manages to force either a red
G
or a blue
H
. The minimum number of rounds for Builder to win the game, regardless of Painter’s strategy, is the on-line Ramsey number
r
~
(
G
,
H
)
. This note focuses on the case when
G
and
H
are stars and paths. In particular, we will prove the upper bound of
r
~
(
S
3
,
P
l
+
1
)
≤
5
l
/
3
+
2
. We will also present an alternative proof for the upper bound of
r
~
(
S
2
=
P
3
,
P
l
+
1
)
=
⌈
5
l
/
4
⌉
.</abstract><cop>Singapore</cop><pub>Springer Singapore</pub><doi>10.1007/s40840-021-01130-x</doi><tpages>11</tpages><orcidid>https://orcid.org/0000-0002-5739-3242</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0126-6705 |
| ispartof | Bulletin of the Malaysian Mathematical Sciences Society, 2021, Vol.44 (5), p.3511-3521 |
| issn | 0126-6705 2180-4206 |
| language | eng |
| recordid | cdi_proquest_journals_2558574718 |
| source | SpringerNature Journals |
| subjects | Apexes Applications of Mathematics Games Graph coloring Graph theory Mathematics Mathematics and Statistics Upper bounds |
| title | A Note on On-Line Ramsey Numbers of Stars and Paths |
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