Maximally resonant polygonal systems
A benzenoid system G is k -resonant if any set F of no more than k disjoint hexagons is a resonant pattern, i.e, G − F has a perfect matching. In 1990’s M. Zheng constructed the 3-resonant benzenoid systems and showed that they are maximally resonant, that is, they are k -resonant for all k ≥ 1 . Re...
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| Veröffentlicht in: | Discrete mathematics 2010-11, Vol.310 (21), p.2790-2800 |
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| creator | Liu, Saihua Zhang, Heping |
| description | A benzenoid system
G
is
k
-resonant if any set
F
of no more than
k
disjoint hexagons is a resonant pattern, i.e,
G
−
F
has a perfect matching. In 1990’s M. Zheng constructed the 3-resonant benzenoid systems and showed that they are maximally resonant, that is, they are
k
-resonant for all
k
≥
1
. Recently, the equivalence of 3-resonance and maximal resonance has been shown to be valid also for coronoid systems, carbon nanotubes, polyhexes in tori and Klein bottles, and fullerene graphs. So our main problem is to investigate the extent of graphs possessing this interesting property. In this paper, by replacing the above hexagons with even faces, we define
k
-resonance of graphs in surfaces, possibly with boundary, in a unified way. Some exceptions exist. For plane polygonal systems tessellated with polygons of even size at least six such that all inner vertices have the same degree three and the others have degree two or three, we show that such 3-resonant polygonal systems are indeed maximally resonant. They can be constructed by gluing and lapping operations on three types of basic graphs. |
| doi_str_mv | 10.1016/j.disc.2010.06.011 |
| format | Article |
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G
is
k
-resonant if any set
F
of no more than
k
disjoint hexagons is a resonant pattern, i.e,
G
−
F
has a perfect matching. In 1990’s M. Zheng constructed the 3-resonant benzenoid systems and showed that they are maximally resonant, that is, they are
k
-resonant for all
k
≥
1
. Recently, the equivalence of 3-resonance and maximal resonance has been shown to be valid also for coronoid systems, carbon nanotubes, polyhexes in tori and Klein bottles, and fullerene graphs. So our main problem is to investigate the extent of graphs possessing this interesting property. In this paper, by replacing the above hexagons with even faces, we define
k
-resonance of graphs in surfaces, possibly with boundary, in a unified way. Some exceptions exist. For plane polygonal systems tessellated with polygons of even size at least six such that all inner vertices have the same degree three and the others have degree two or three, we show that such 3-resonant polygonal systems are indeed maximally resonant. They can be constructed by gluing and lapping operations on three types of basic graphs.</description><identifier>ISSN: 0012-365X</identifier><identifier>EISSN: 1872-681X</identifier><identifier>DOI: 10.1016/j.disc.2010.06.011</identifier><identifier>CODEN: DSMHA4</identifier><language>eng</language><publisher>Kidlington: Elsevier B.V</publisher><subject>[formula omitted]-resonance ; Algebra ; Applied sciences ; Benzenoids ; Boundaries ; Combinatorics ; Combinatorics. Ordered structures ; Computer science; control theory; systems ; Construction ; Exact sciences and technology ; Graph theory ; Graphs ; Hexagons ; Information retrieval. Graph ; Inner dual ; Lapping ; Mathematical analysis ; Mathematics ; Partial differential equations ; Perfect matching ; Polygonal system ; Polygons ; Sciences and techniques of general use ; Theoretical computing</subject><ispartof>Discrete mathematics, 2010-11, Vol.310 (21), p.2790-2800</ispartof><rights>2010</rights><rights>2015 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-d4ca5b0fff4b48051229b0180b7ac132769dde8a1ad26005408286450d67a1513</citedby><cites>FETCH-LOGICAL-c363t-d4ca5b0fff4b48051229b0180b7ac132769dde8a1ad26005408286450d67a1513</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.disc.2010.06.011$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=23212828$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Liu, Saihua</creatorcontrib><creatorcontrib>Zhang, Heping</creatorcontrib><title>Maximally resonant polygonal systems</title><title>Discrete mathematics</title><description>A benzenoid system
G
is
k
-resonant if any set
F
of no more than
k
disjoint hexagons is a resonant pattern, i.e,
G
−
F
has a perfect matching. In 1990’s M. Zheng constructed the 3-resonant benzenoid systems and showed that they are maximally resonant, that is, they are
k
-resonant for all
k
≥
1
. Recently, the equivalence of 3-resonance and maximal resonance has been shown to be valid also for coronoid systems, carbon nanotubes, polyhexes in tori and Klein bottles, and fullerene graphs. So our main problem is to investigate the extent of graphs possessing this interesting property. In this paper, by replacing the above hexagons with even faces, we define
k
-resonance of graphs in surfaces, possibly with boundary, in a unified way. Some exceptions exist. For plane polygonal systems tessellated with polygons of even size at least six such that all inner vertices have the same degree three and the others have degree two or three, we show that such 3-resonant polygonal systems are indeed maximally resonant. They can be constructed by gluing and lapping operations on three types of basic graphs.</description><subject>[formula omitted]-resonance</subject><subject>Algebra</subject><subject>Applied sciences</subject><subject>Benzenoids</subject><subject>Boundaries</subject><subject>Combinatorics</subject><subject>Combinatorics. Ordered structures</subject><subject>Computer science; control theory; systems</subject><subject>Construction</subject><subject>Exact sciences and technology</subject><subject>Graph theory</subject><subject>Graphs</subject><subject>Hexagons</subject><subject>Information retrieval. Graph</subject><subject>Inner dual</subject><subject>Lapping</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Partial differential equations</subject><subject>Perfect matching</subject><subject>Polygonal system</subject><subject>Polygons</subject><subject>Sciences and techniques of general use</subject><subject>Theoretical computing</subject><issn>0012-365X</issn><issn>1872-681X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2010</creationdate><recordtype>article</recordtype><recordid>eNp9kMtKAzEUhoMoWKsv4KoLBTcznpPMZFJwI8UbVNwodBfSJCNT0pmaMxXn7U1pcenqXPjO5f8Zu0TIEVDernLXkM05pAbIHBCP2AhVxTOpcHHMRgDIMyHLxSk7I1pBqqVQI3b1an6atQlhmERPXWvafrLpwvCZ0jChgXq_pnN2UptA_uIQx-zj8eF99pzN355eZvfzzAop-swV1pRLqOu6WBYKSuR8ugRUsKyMRcErOXXOK4PGcQlQFqC4kkUJTlYGSxRjdrPfu4nd19ZTr9dJlQ_BtL7bkkZZIRdlIhPK96iNHVH0td7EpCMOGkHvLNErvbNE7yzRIHWyJA1dH_YbsibU0bS2ob9JLjjy9FLi7vacT2K_Gx812ca31rsmettr1zX_nfkFLzt1RQ</recordid><startdate>20101106</startdate><enddate>20101106</enddate><creator>Liu, Saihua</creator><creator>Zhang, Heping</creator><general>Elsevier B.V</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20101106</creationdate><title>Maximally resonant polygonal systems</title><author>Liu, Saihua ; Zhang, Heping</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-d4ca5b0fff4b48051229b0180b7ac132769dde8a1ad26005408286450d67a1513</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2010</creationdate><topic>[formula omitted]-resonance</topic><topic>Algebra</topic><topic>Applied sciences</topic><topic>Benzenoids</topic><topic>Boundaries</topic><topic>Combinatorics</topic><topic>Combinatorics. Ordered structures</topic><topic>Computer science; control theory; systems</topic><topic>Construction</topic><topic>Exact sciences and technology</topic><topic>Graph theory</topic><topic>Graphs</topic><topic>Hexagons</topic><topic>Information retrieval. Graph</topic><topic>Inner dual</topic><topic>Lapping</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Partial differential equations</topic><topic>Perfect matching</topic><topic>Polygonal system</topic><topic>Polygons</topic><topic>Sciences and techniques of general use</topic><topic>Theoretical computing</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Liu, Saihua</creatorcontrib><creatorcontrib>Zhang, Heping</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</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>Discrete mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Liu, Saihua</au><au>Zhang, Heping</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Maximally resonant polygonal systems</atitle><jtitle>Discrete mathematics</jtitle><date>2010-11-06</date><risdate>2010</risdate><volume>310</volume><issue>21</issue><spage>2790</spage><epage>2800</epage><pages>2790-2800</pages><issn>0012-365X</issn><eissn>1872-681X</eissn><coden>DSMHA4</coden><abstract>A benzenoid system
G
is
k
-resonant if any set
F
of no more than
k
disjoint hexagons is a resonant pattern, i.e,
G
−
F
has a perfect matching. In 1990’s M. Zheng constructed the 3-resonant benzenoid systems and showed that they are maximally resonant, that is, they are
k
-resonant for all
k
≥
1
. Recently, the equivalence of 3-resonance and maximal resonance has been shown to be valid also for coronoid systems, carbon nanotubes, polyhexes in tori and Klein bottles, and fullerene graphs. So our main problem is to investigate the extent of graphs possessing this interesting property. In this paper, by replacing the above hexagons with even faces, we define
k
-resonance of graphs in surfaces, possibly with boundary, in a unified way. Some exceptions exist. For plane polygonal systems tessellated with polygons of even size at least six such that all inner vertices have the same degree three and the others have degree two or three, we show that such 3-resonant polygonal systems are indeed maximally resonant. They can be constructed by gluing and lapping operations on three types of basic graphs.</abstract><cop>Kidlington</cop><pub>Elsevier B.V</pub><doi>10.1016/j.disc.2010.06.011</doi><tpages>11</tpages><oa>free_for_read</oa></addata></record> |
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| source | Access via ScienceDirect (Elsevier); Free E-Journal (出版社公開部分のみ） |
| subjects | [formula omitted]-resonance Algebra Applied sciences Benzenoids Boundaries Combinatorics Combinatorics. Ordered structures Computer science control theory systems Construction Exact sciences and technology Graph theory Graphs Hexagons Information retrieval. Graph Inner dual Lapping Mathematical analysis Mathematics Partial differential equations Perfect matching Polygonal system Polygons Sciences and techniques of general use Theoretical computing |
| title | Maximally resonant polygonal systems |
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