On 0-Rotatable Graceful Caterpillars
An injection f : V ( T ) → { 0 , … , | E ( T ) | } of a tree T is a graceful labelling if { | f ( u ) - f ( v ) | : u v ∈ E ( T ) } = { 1 , … , | E ( T ) | } . Tree T is 0-rotatable if, for any v ∈ V ( T ) , there exists a graceful labelling f of T such that f ( v ) = 0 . In this work, the following...
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| Veröffentlicht in: | Graphs and combinatorics 2020-11, Vol.36 (6), p.1655-1673 |
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| container_title | Graphs and combinatorics |
| container_volume | 36 |
| creator | Luiz, Atílio G. Campos, C. N. Richter, R. Bruce |
| description | An injection
f
:
V
(
T
)
→
{
0
,
…
,
|
E
(
T
)
|
}
of a tree
T
is a graceful labelling if
{
|
f
(
u
)
-
f
(
v
)
|
:
u
v
∈
E
(
T
)
}
=
{
1
,
…
,
|
E
(
T
)
|
}
. Tree
T
is 0-rotatable if, for any
v
∈
V
(
T
)
, there exists a graceful labelling
f
of
T
such that
f
(
v
)
=
0
. In this work, the following families of caterpillars are proved to be 0-rotatable: caterpillars with a perfect matching; caterpillars obtained by linking one leaf of the star
K
1
,
s
-
1
to a leaf of a path
P
n
with
n
≥
3
and
s
≥
⌈
n
2
⌉
; caterpillars with diameter five or six; and caterpillars
T
with
diam
(
T
)
≥
7
such that, for every non-leaf vertex
v
∈
V
(
T
)
, the number of leaves adjacent to
v
is even and is at least
2
+
2
(
(
diam
(
T
)
-
1
)
mod
2
)
. These results reinforce the conjecture that all caterpillars with diameter at least five are 0-rotatable. |
| doi_str_mv | 10.1007/s00373-020-02226-0 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2451546112</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2451546112</sourcerecordid><originalsourceid>FETCH-LOGICAL-c270t-ae09419bdebc7c682e4ada8e954586d216aa89b78659bcd8c91b0d2e2c3d01a3</originalsourceid><addsrcrecordid>eNp9kEFLxDAQhYMoWFf_gKeCXqMzaZI2R1l0FRYWZO9hmqSyS21r0h7891YrePMwvMv73sDH2DXCHQKU9wmgKAsOAuYTQnM4YRnKQnFlUJ6yDAwiB0Rzzi5SOgKAQgkZu911OfDXfqSR6jbkm0guNFObr2kMcTi0LcV0yc4aalO4-s0V2z897tfPfLvbvKwfttyJEkZOAYxEU_tQu9LpSgRJnqpglFSV9gI1UWXqstLK1M5XzmANXgThCg9IxYrdLLND7D-mkEZ77KfYzR-tkAqV1Ihiboml5WKfUgyNHeLhneKnRbDfMuwiw84y7I8MCzNULFCay91biH_T_1Bfb9lf8w</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2451546112</pqid></control><display><type>article</type><title>On 0-Rotatable Graceful Caterpillars</title><source>Springer Online Journals Complete</source><creator>Luiz, Atílio G. ; Campos, C. N. ; Richter, R. Bruce</creator><creatorcontrib>Luiz, Atílio G. ; Campos, C. N. ; Richter, R. Bruce</creatorcontrib><description>An injection
f
:
V
(
T
)
→
{
0
,
…
,
|
E
(
T
)
|
}
of a tree
T
is a graceful labelling if
{
|
f
(
u
)
-
f
(
v
)
|
:
u
v
∈
E
(
T
)
}
=
{
1
,
…
,
|
E
(
T
)
|
}
. Tree
T
is 0-rotatable if, for any
v
∈
V
(
T
)
, there exists a graceful labelling
f
of
T
such that
f
(
v
)
=
0
. In this work, the following families of caterpillars are proved to be 0-rotatable: caterpillars with a perfect matching; caterpillars obtained by linking one leaf of the star
K
1
,
s
-
1
to a leaf of a path
P
n
with
n
≥
3
and
s
≥
⌈
n
2
⌉
; caterpillars with diameter five or six; and caterpillars
T
with
diam
(
T
)
≥
7
such that, for every non-leaf vertex
v
∈
V
(
T
)
, the number of leaves adjacent to
v
is even and is at least
2
+
2
(
(
diam
(
T
)
-
1
)
mod
2
)
. These results reinforce the conjecture that all caterpillars with diameter at least five are 0-rotatable.</description><identifier>ISSN: 0911-0119</identifier><identifier>EISSN: 1435-5914</identifier><identifier>DOI: 10.1007/s00373-020-02226-0</identifier><language>eng</language><publisher>Tokyo: Springer Japan</publisher><subject>Caterpillars ; Combinatorics ; Engineering Design ; Labeling ; Mathematics ; Mathematics and Statistics ; Original Paper</subject><ispartof>Graphs and combinatorics, 2020-11, Vol.36 (6), p.1655-1673</ispartof><rights>Springer Japan KK, part of Springer Nature 2020</rights><rights>Springer Japan KK, part of Springer Nature 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-ae09419bdebc7c682e4ada8e954586d216aa89b78659bcd8c91b0d2e2c3d01a3</cites><orcidid>0000-0002-6177-403X</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/s00373-020-02226-0$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00373-020-02226-0$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Luiz, Atílio G.</creatorcontrib><creatorcontrib>Campos, C. N.</creatorcontrib><creatorcontrib>Richter, R. Bruce</creatorcontrib><title>On 0-Rotatable Graceful Caterpillars</title><title>Graphs and combinatorics</title><addtitle>Graphs and Combinatorics</addtitle><description>An injection
f
:
V
(
T
)
→
{
0
,
…
,
|
E
(
T
)
|
}
of a tree
T
is a graceful labelling if
{
|
f
(
u
)
-
f
(
v
)
|
:
u
v
∈
E
(
T
)
}
=
{
1
,
…
,
|
E
(
T
)
|
}
. Tree
T
is 0-rotatable if, for any
v
∈
V
(
T
)
, there exists a graceful labelling
f
of
T
such that
f
(
v
)
=
0
. In this work, the following families of caterpillars are proved to be 0-rotatable: caterpillars with a perfect matching; caterpillars obtained by linking one leaf of the star
K
1
,
s
-
1
to a leaf of a path
P
n
with
n
≥
3
and
s
≥
⌈
n
2
⌉
; caterpillars with diameter five or six; and caterpillars
T
with
diam
(
T
)
≥
7
such that, for every non-leaf vertex
v
∈
V
(
T
)
, the number of leaves adjacent to
v
is even and is at least
2
+
2
(
(
diam
(
T
)
-
1
)
mod
2
)
. These results reinforce the conjecture that all caterpillars with diameter at least five are 0-rotatable.</description><subject>Caterpillars</subject><subject>Combinatorics</subject><subject>Engineering Design</subject><subject>Labeling</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><recordid>eNp9kEFLxDAQhYMoWFf_gKeCXqMzaZI2R1l0FRYWZO9hmqSyS21r0h7891YrePMwvMv73sDH2DXCHQKU9wmgKAsOAuYTQnM4YRnKQnFlUJ6yDAwiB0Rzzi5SOgKAQgkZu911OfDXfqSR6jbkm0guNFObr2kMcTi0LcV0yc4aalO4-s0V2z897tfPfLvbvKwfttyJEkZOAYxEU_tQu9LpSgRJnqpglFSV9gI1UWXqstLK1M5XzmANXgThCg9IxYrdLLND7D-mkEZ77KfYzR-tkAqV1Ihiboml5WKfUgyNHeLhneKnRbDfMuwiw84y7I8MCzNULFCay91biH_T_1Bfb9lf8w</recordid><startdate>20201101</startdate><enddate>20201101</enddate><creator>Luiz, Atílio G.</creator><creator>Campos, C. N.</creator><creator>Richter, R. Bruce</creator><general>Springer Japan</general><general>Springer Nature B.V</general><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><orcidid>https://orcid.org/0000-0002-6177-403X</orcidid></search><sort><creationdate>20201101</creationdate><title>On 0-Rotatable Graceful Caterpillars</title><author>Luiz, Atílio G. ; Campos, C. N. ; Richter, R. Bruce</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-ae09419bdebc7c682e4ada8e954586d216aa89b78659bcd8c91b0d2e2c3d01a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Caterpillars</topic><topic>Combinatorics</topic><topic>Engineering Design</topic><topic>Labeling</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Luiz, Atílio G.</creatorcontrib><creatorcontrib>Campos, C. N.</creatorcontrib><creatorcontrib>Richter, R. Bruce</creatorcontrib><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>Luiz, Atílio G.</au><au>Campos, C. N.</au><au>Richter, R. Bruce</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On 0-Rotatable Graceful Caterpillars</atitle><jtitle>Graphs and combinatorics</jtitle><stitle>Graphs and Combinatorics</stitle><date>2020-11-01</date><risdate>2020</risdate><volume>36</volume><issue>6</issue><spage>1655</spage><epage>1673</epage><pages>1655-1673</pages><issn>0911-0119</issn><eissn>1435-5914</eissn><abstract>An injection
f
:
V
(
T
)
→
{
0
,
…
,
|
E
(
T
)
|
}
of a tree
T
is a graceful labelling if
{
|
f
(
u
)
-
f
(
v
)
|
:
u
v
∈
E
(
T
)
}
=
{
1
,
…
,
|
E
(
T
)
|
}
. Tree
T
is 0-rotatable if, for any
v
∈
V
(
T
)
, there exists a graceful labelling
f
of
T
such that
f
(
v
)
=
0
. In this work, the following families of caterpillars are proved to be 0-rotatable: caterpillars with a perfect matching; caterpillars obtained by linking one leaf of the star
K
1
,
s
-
1
to a leaf of a path
P
n
with
n
≥
3
and
s
≥
⌈
n
2
⌉
; caterpillars with diameter five or six; and caterpillars
T
with
diam
(
T
)
≥
7
such that, for every non-leaf vertex
v
∈
V
(
T
)
, the number of leaves adjacent to
v
is even and is at least
2
+
2
(
(
diam
(
T
)
-
1
)
mod
2
)
. These results reinforce the conjecture that all caterpillars with diameter at least five are 0-rotatable.</abstract><cop>Tokyo</cop><pub>Springer Japan</pub><doi>10.1007/s00373-020-02226-0</doi><tpages>19</tpages><orcidid>https://orcid.org/0000-0002-6177-403X</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0911-0119 |
| ispartof | Graphs and combinatorics, 2020-11, Vol.36 (6), p.1655-1673 |
| issn | 0911-0119 1435-5914 |
| language | eng |
| recordid | cdi_proquest_journals_2451546112 |
| source | Springer Online Journals Complete |
| subjects | Caterpillars Combinatorics Engineering Design Labeling Mathematics Mathematics and Statistics Original Paper |
| title | On 0-Rotatable Graceful Caterpillars |
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