Some Results on the 3-Vertex-Rainbow Index of a Graph
Let G be a nontrivial connected graph with a vertex-coloring c : V ( G ) → { 1 , 2 , … , q } , q ∈ N . For a set S ⊆ V ( G ) and | S | ≥ 2 , a subtree T of G satisfying S ⊆ V ( T ) is said to be an S -Steiner tree or simply S -tree. The S -tree T is called a vertex-rainbow S -tree if the vertices of...
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| Veröffentlicht in: | Bulletin of the Malaysian Mathematical Sciences Society 2021-03, Vol.44 (2), p.1015-1034 |
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| creator | Ma, Yingbin Zhu, Wenhan |
| description | Let
G
be a nontrivial connected graph with a vertex-coloring
c
:
V
(
G
)
→
{
1
,
2
,
…
,
q
}
,
q
∈
N
. For a set
S
⊆
V
(
G
)
and
|
S
|
≥
2
, a subtree
T
of
G
satisfying
S
⊆
V
(
T
)
is said to be an
S
-Steiner tree or simply
S
-tree. The
S
-tree
T
is called a vertex-rainbow
S
-tree if the vertices of
V
(
T
)
\
S
have distinct colors. Let
k
be a fixed integer with
2
≤
k
≤
|
V
(
G
)
|
, if every
k
-subset
S
of
V
(
G
) has a vertex-rainbow
S
-tree, then
G
is said to be vertex-rainbow
k
-tree connected. The
k
-vertex-rainbow index of
G
, denoted by
r
v
x
k
(
G
)
, is the minimum number of colors that are needed in order to make
G
vertex-rainbow
k
-tree connected. In this paper, we study the 3-vertex-rainbow index of unicyclic graphs and complementary graphs, respectively. |
| doi_str_mv | 10.1007/s40840-020-00992-x |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2490667139</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2490667139</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-b530ef708a36bee402002677a07e1c0dbf947dd69d9a6f3ec4f59be2e6e99d033</originalsourceid><addsrcrecordid>eNp9kM1LAzEQxYMoWGr_AU8Bz9HJxybNUYrWQkGoH9eQ3Z21Le2mJltc_3ujK3hzYJjL7703PEIuOVxzAHOTFEwVMBB5wVrB-hMyEnwKTAnQp2QEXGimDRTnZJLSFvIUWmjBR6R4CnukK0zHXZdoaGm3RirZK8YOe7bym7YMH3TR1tjT0FBP59Ef1hfkrPG7hJPfOyYv93fPswe2fJwvZrdLVkluO1YWErAxMPVSl4gqfwhCG-PBIK-gLhurTF1rW1uvG4mVagpbokCN1tYg5ZhcDb6HGN6PmDq3DcfY5kgnlAWtDZc2U2KgqhhSiti4Q9zsffx0HNx3Q25oyOV899OQ67NIDqKU4fYN45_1P6ovRAFnLg</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2490667139</pqid></control><display><type>article</type><title>Some Results on the 3-Vertex-Rainbow Index of a Graph</title><source>Springer Nature - Complete Springer Journals</source><creator>Ma, Yingbin ; Zhu, Wenhan</creator><creatorcontrib>Ma, Yingbin ; Zhu, Wenhan</creatorcontrib><description>Let
G
be a nontrivial connected graph with a vertex-coloring
c
:
V
(
G
)
→
{
1
,
2
,
…
,
q
}
,
q
∈
N
. For a set
S
⊆
V
(
G
)
and
|
S
|
≥
2
, a subtree
T
of
G
satisfying
S
⊆
V
(
T
)
is said to be an
S
-Steiner tree or simply
S
-tree. The
S
-tree
T
is called a vertex-rainbow
S
-tree if the vertices of
V
(
T
)
\
S
have distinct colors. Let
k
be a fixed integer with
2
≤
k
≤
|
V
(
G
)
|
, if every
k
-subset
S
of
V
(
G
) has a vertex-rainbow
S
-tree, then
G
is said to be vertex-rainbow
k
-tree connected. The
k
-vertex-rainbow index of
G
, denoted by
r
v
x
k
(
G
)
, is the minimum number of colors that are needed in order to make
G
vertex-rainbow
k
-tree connected. In this paper, we study the 3-vertex-rainbow index of unicyclic graphs and complementary graphs, respectively.</description><identifier>ISSN: 0126-6705</identifier><identifier>EISSN: 2180-4206</identifier><identifier>DOI: 10.1007/s40840-020-00992-x</identifier><language>eng</language><publisher>Singapore: Springer Singapore</publisher><subject>Apexes ; Applications of Mathematics ; Graph coloring ; Graphs ; Mathematics ; Mathematics and Statistics</subject><ispartof>Bulletin of the Malaysian Mathematical Sciences Society, 2021-03, Vol.44 (2), p.1015-1034</ispartof><rights>Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020</rights><rights>Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-b530ef708a36bee402002677a07e1c0dbf947dd69d9a6f3ec4f59be2e6e99d033</citedby><cites>FETCH-LOGICAL-c319t-b530ef708a36bee402002677a07e1c0dbf947dd69d9a6f3ec4f59be2e6e99d033</cites><orcidid>0000-0002-0380-9468</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/s40840-020-00992-x$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s40840-020-00992-x$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,778,782,27911,27912,41475,42544,51306</link.rule.ids></links><search><creatorcontrib>Ma, Yingbin</creatorcontrib><creatorcontrib>Zhu, Wenhan</creatorcontrib><title>Some Results on the 3-Vertex-Rainbow Index of a Graph</title><title>Bulletin of the Malaysian Mathematical Sciences Society</title><addtitle>Bull. Malays. Math. Sci. Soc</addtitle><description>Let
G
be a nontrivial connected graph with a vertex-coloring
c
:
V
(
G
)
→
{
1
,
2
,
…
,
q
}
,
q
∈
N
. For a set
S
⊆
V
(
G
)
and
|
S
|
≥
2
, a subtree
T
of
G
satisfying
S
⊆
V
(
T
)
is said to be an
S
-Steiner tree or simply
S
-tree. The
S
-tree
T
is called a vertex-rainbow
S
-tree if the vertices of
V
(
T
)
\
S
have distinct colors. Let
k
be a fixed integer with
2
≤
k
≤
|
V
(
G
)
|
, if every
k
-subset
S
of
V
(
G
) has a vertex-rainbow
S
-tree, then
G
is said to be vertex-rainbow
k
-tree connected. The
k
-vertex-rainbow index of
G
, denoted by
r
v
x
k
(
G
)
, is the minimum number of colors that are needed in order to make
G
vertex-rainbow
k
-tree connected. In this paper, we study the 3-vertex-rainbow index of unicyclic graphs and complementary graphs, respectively.</description><subject>Apexes</subject><subject>Applications of Mathematics</subject><subject>Graph coloring</subject><subject>Graphs</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0126-6705</issn><issn>2180-4206</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kM1LAzEQxYMoWGr_AU8Bz9HJxybNUYrWQkGoH9eQ3Z21Le2mJltc_3ujK3hzYJjL7703PEIuOVxzAHOTFEwVMBB5wVrB-hMyEnwKTAnQp2QEXGimDRTnZJLSFvIUWmjBR6R4CnukK0zHXZdoaGm3RirZK8YOe7bym7YMH3TR1tjT0FBP59Ef1hfkrPG7hJPfOyYv93fPswe2fJwvZrdLVkluO1YWErAxMPVSl4gqfwhCG-PBIK-gLhurTF1rW1uvG4mVagpbokCN1tYg5ZhcDb6HGN6PmDq3DcfY5kgnlAWtDZc2U2KgqhhSiti4Q9zsffx0HNx3Q25oyOV899OQ67NIDqKU4fYN45_1P6ovRAFnLg</recordid><startdate>20210301</startdate><enddate>20210301</enddate><creator>Ma, Yingbin</creator><creator>Zhu, Wenhan</creator><general>Springer Singapore</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-0380-9468</orcidid></search><sort><creationdate>20210301</creationdate><title>Some Results on the 3-Vertex-Rainbow Index of a Graph</title><author>Ma, Yingbin ; Zhu, Wenhan</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-b530ef708a36bee402002677a07e1c0dbf947dd69d9a6f3ec4f59be2e6e99d033</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Apexes</topic><topic>Applications of Mathematics</topic><topic>Graph coloring</topic><topic>Graphs</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ma, Yingbin</creatorcontrib><creatorcontrib>Zhu, Wenhan</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>Ma, Yingbin</au><au>Zhu, Wenhan</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Some Results on the 3-Vertex-Rainbow Index of a Graph</atitle><jtitle>Bulletin of the Malaysian Mathematical Sciences Society</jtitle><stitle>Bull. Malays. Math. Sci. Soc</stitle><date>2021-03-01</date><risdate>2021</risdate><volume>44</volume><issue>2</issue><spage>1015</spage><epage>1034</epage><pages>1015-1034</pages><issn>0126-6705</issn><eissn>2180-4206</eissn><abstract>Let
G
be a nontrivial connected graph with a vertex-coloring
c
:
V
(
G
)
→
{
1
,
2
,
…
,
q
}
,
q
∈
N
. For a set
S
⊆
V
(
G
)
and
|
S
|
≥
2
, a subtree
T
of
G
satisfying
S
⊆
V
(
T
)
is said to be an
S
-Steiner tree or simply
S
-tree. The
S
-tree
T
is called a vertex-rainbow
S
-tree if the vertices of
V
(
T
)
\
S
have distinct colors. Let
k
be a fixed integer with
2
≤
k
≤
|
V
(
G
)
|
, if every
k
-subset
S
of
V
(
G
) has a vertex-rainbow
S
-tree, then
G
is said to be vertex-rainbow
k
-tree connected. The
k
-vertex-rainbow index of
G
, denoted by
r
v
x
k
(
G
)
, is the minimum number of colors that are needed in order to make
G
vertex-rainbow
k
-tree connected. In this paper, we study the 3-vertex-rainbow index of unicyclic graphs and complementary graphs, respectively.</abstract><cop>Singapore</cop><pub>Springer Singapore</pub><doi>10.1007/s40840-020-00992-x</doi><tpages>20</tpages><orcidid>https://orcid.org/0000-0002-0380-9468</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0126-6705 |
| ispartof | Bulletin of the Malaysian Mathematical Sciences Society, 2021-03, Vol.44 (2), p.1015-1034 |
| issn | 0126-6705 2180-4206 |
| language | eng |
| recordid | cdi_proquest_journals_2490667139 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Apexes Applications of Mathematics Graph coloring Graphs Mathematics Mathematics and Statistics |
| title | Some Results on the 3-Vertex-Rainbow Index of a Graph |
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