Optimal Pebbling Number of the Square Grid
A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number π opt is the smallest number m needed to guaran...
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Veröffentlicht in: | Graphs and combinatorics 2020-05, Vol.36 (3), p.803-829 |
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creator | Győri, Ervin Katona, Gyula Y. Papp, László F. |
description | A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number
π
opt
is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph
P
n
□
P
m
was investigated in several papers (Bunde et al. in J Graph Theory 57(3):215–238, 2008; Xue and Yerger in Graphs Combin 32(3):1229–1247, 2016; Győri et al. in Period Polytech Electr Eng Comput Sci 61(2):217–223 2017). In this paper, we present a new method using some recent ideas to give a lower bound on
π
opt
. We apply this technique to prove that
π
opt
(
P
n
□
P
m
)
≥
2
13
n
m
. Our method also gives a new proof for
π
opt
(
P
n
)
=
π
opt
(
C
n
)
=
2
n
3
. |
doi_str_mv | 10.1007/s00373-020-02154-z |
format | Article |
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π
opt
is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph
P
n
□
P
m
was investigated in several papers (Bunde et al. in J Graph Theory 57(3):215–238, 2008; Xue and Yerger in Graphs Combin 32(3):1229–1247, 2016; Győri et al. in Period Polytech Electr Eng Comput Sci 61(2):217–223 2017). In this paper, we present a new method using some recent ideas to give a lower bound on
π
opt
. We apply this technique to prove that
π
opt
(
P
n
□
P
m
)
≥
2
13
n
m
. Our method also gives a new proof for
π
opt
(
P
n
)
=
π
opt
(
C
n
)
=
2
n
3
.</description><identifier>ISSN: 0911-0119</identifier><identifier>EISSN: 1435-5914</identifier><identifier>DOI: 10.1007/s00373-020-02154-z</identifier><language>eng</language><publisher>Tokyo: Springer Japan</publisher><subject>Combinatorics ; Engineering Design ; Graph theory ; Lower bounds ; Mathematics ; Mathematics and Statistics ; Original Paper</subject><ispartof>Graphs and combinatorics, 2020-05, Vol.36 (3), p.803-829</ispartof><rights>The Author(s) 2020</rights><rights>The Author(s) 2020. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-b7c119f54e1494e64d27007b92d3d6d65650cfb21aca2532a927f6fcb530b9973</citedby><cites>FETCH-LOGICAL-c363t-b7c119f54e1494e64d27007b92d3d6d65650cfb21aca2532a927f6fcb530b9973</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00373-020-02154-z$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00373-020-02154-z$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Győri, Ervin</creatorcontrib><creatorcontrib>Katona, Gyula Y.</creatorcontrib><creatorcontrib>Papp, László F.</creatorcontrib><title>Optimal Pebbling Number of the Square Grid</title><title>Graphs and combinatorics</title><addtitle>Graphs and Combinatorics</addtitle><description>A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number
π
opt
is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph
P
n
□
P
m
was investigated in several papers (Bunde et al. in J Graph Theory 57(3):215–238, 2008; Xue and Yerger in Graphs Combin 32(3):1229–1247, 2016; Győri et al. in Period Polytech Electr Eng Comput Sci 61(2):217–223 2017). In this paper, we present a new method using some recent ideas to give a lower bound on
π
opt
. We apply this technique to prove that
π
opt
(
P
n
□
P
m
)
≥
2
13
n
m
. Our method also gives a new proof for
π
opt
(
P
n
)
=
π
opt
(
C
n
)
=
2
n
3
.</description><subject>Combinatorics</subject><subject>Engineering Design</subject><subject>Graph theory</subject><subject>Lower bounds</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><issn>0911-0119</issn><issn>1435-5914</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kE1LAzEQhoMouFb_gKcFb0J0Jp_kKEWrUKygnsNmN6kt226b7B7srze6gjcPw1zej5mHkEuEGwTQtwmAa06BQR6Ugh6OSIGCSyoNimNSgEGkgGhOyVlKawCQKKAg14tdv9pUbfninWtX22X5PGycj2UXyv7Dl6_7oYq-nMVVc05OQtUmf_G7J-T94f5t-kjni9nT9G5Oa654T52uc02QwqMwwivRMJ1PdIY1vFGNkkpCHRzDqq6Y5KwyTAcVaic5OGM0n5CrMXcXu_3gU2_X3RC3udIybrhCrtFkFRtVdexSij7YXcyPxE-LYL-Z2JGJzUzsDxN7yCY-mlIWb5c-_kX_4_oCMNViLg</recordid><startdate>20200501</startdate><enddate>20200501</enddate><creator>Győri, Ervin</creator><creator>Katona, Gyula Y.</creator><creator>Papp, László F.</creator><general>Springer Japan</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20200501</creationdate><title>Optimal Pebbling Number of the Square Grid</title><author>Győri, Ervin ; Katona, Gyula Y. ; Papp, László F.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-b7c119f54e1494e64d27007b92d3d6d65650cfb21aca2532a927f6fcb530b9973</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Combinatorics</topic><topic>Engineering Design</topic><topic>Graph theory</topic><topic>Lower bounds</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Győri, Ervin</creatorcontrib><creatorcontrib>Katona, Gyula Y.</creatorcontrib><creatorcontrib>Papp, László F.</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Graphs and combinatorics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Győri, Ervin</au><au>Katona, Gyula Y.</au><au>Papp, László F.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Optimal Pebbling Number of the Square Grid</atitle><jtitle>Graphs and combinatorics</jtitle><stitle>Graphs and Combinatorics</stitle><date>2020-05-01</date><risdate>2020</risdate><volume>36</volume><issue>3</issue><spage>803</spage><epage>829</epage><pages>803-829</pages><issn>0911-0119</issn><eissn>1435-5914</eissn><abstract>A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number
π
opt
is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph
P
n
□
P
m
was investigated in several papers (Bunde et al. in J Graph Theory 57(3):215–238, 2008; Xue and Yerger in Graphs Combin 32(3):1229–1247, 2016; Győri et al. in Period Polytech Electr Eng Comput Sci 61(2):217–223 2017). In this paper, we present a new method using some recent ideas to give a lower bound on
π
opt
. We apply this technique to prove that
π
opt
(
P
n
□
P
m
)
≥
2
13
n
m
. Our method also gives a new proof for
π
opt
(
P
n
)
=
π
opt
(
C
n
)
=
2
n
3
.</abstract><cop>Tokyo</cop><pub>Springer Japan</pub><doi>10.1007/s00373-020-02154-z</doi><tpages>27</tpages><oa>free_for_read</oa></addata></record> |
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subjects | Combinatorics Engineering Design Graph theory Lower bounds Mathematics Mathematics and Statistics Original Paper |
title | Optimal Pebbling Number of the Square Grid |
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