Graphic sequences with a realization containing a generalized friendship graph
Gould, Jacobson and Lehel [R.J. Gould, M.S. Jacobson, J. Lehel, Potentially G-graphical degree sequences, in: Y. Alavi, et al. (Eds.), Combinatorics, Graph Theory and Algorithms, vol. I, New Issues Press, Kalamazoo, MI, 1999, pp. 451–460] considered a variation of the classical Turán-type extremal p...
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| Veröffentlicht in: | Discrete mathematics Jg. 308; H. 24; S. 6226 - 6232 |
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| Sprache: | Englisch |
| Veröffentlicht: |
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Elsevier B.V
28.12.2008
Elsevier |
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| ISSN: | 0012-365X, 1872-681X |
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| Abstract | Gould, Jacobson and Lehel [R.J. Gould, M.S. Jacobson, J. Lehel, Potentially G-graphical degree sequences, in: Y. Alavi, et al. (Eds.), Combinatorics, Graph Theory and Algorithms, vol. I, New Issues Press, Kalamazoo, MI, 1999, pp. 451–460] considered a variation of the classical Turán-type extremal problems as follows: for any simple graph
H
, determine the smallest even integer
σ
(
H
,
n
)
such that every
n
-term graphic sequence
π
=
(
d
1
,
d
2
,
…
,
d
n
)
with term sum
σ
(
π
)
=
d
1
+
d
2
+
⋯
+
d
n
≥
σ
(
H
,
n
)
has a realization
G
containing
H
as a subgraph. Let
F
t
,
r
,
k
denote the generalized friendship graph on
k
t
−
k
r
+
r
vertices, that is, the graph of
k
copies of
K
t
meeting in a common
r
set, where
K
t
is the complete graph on
t
vertices and
0
≤
r
≤
t
. In this paper, we determine
σ
(
F
t
,
r
,
k
,
n
)
for
k
≥
2
,
t
≥
3
,
1
≤
r
≤
t
−
2
and
n
sufficiently large. |
|---|---|
| AbstractList | Gould, Jacobson and Lehel [R.J. Gould, M.S. Jacobson, J. Lehel, Potentially G-graphical degree sequences, in: Y. Alavi, et al. (Eds.), Combinatorics, Graph Theory and Algorithms, vol. I, New Issues Press, Kalamazoo, MI, 1999, pp. 451–460] considered a variation of the classical Turán-type extremal problems as follows: for any simple graph
H
, determine the smallest even integer
σ
(
H
,
n
)
such that every
n
-term graphic sequence
π
=
(
d
1
,
d
2
,
…
,
d
n
)
with term sum
σ
(
π
)
=
d
1
+
d
2
+
⋯
+
d
n
≥
σ
(
H
,
n
)
has a realization
G
containing
H
as a subgraph. Let
F
t
,
r
,
k
denote the generalized friendship graph on
k
t
−
k
r
+
r
vertices, that is, the graph of
k
copies of
K
t
meeting in a common
r
set, where
K
t
is the complete graph on
t
vertices and
0
≤
r
≤
t
. In this paper, we determine
σ
(
F
t
,
r
,
k
,
n
)
for
k
≥
2
,
t
≥
3
,
1
≤
r
≤
t
−
2
and
n
sufficiently large. |
| Author | Schmitt, John R. Chen, Gang Yin, Jian-Hua |
| Author_xml | – sequence: 1 givenname: Jian-Hua surname: Yin fullname: Yin, Jian-Hua email: yinjh@ustc.edu organization: Department of Applied Mathematics, College of Information Science and Technology, Hainan University, Haikou 570228, PR China – sequence: 2 givenname: Gang surname: Chen fullname: Chen, Gang organization: Department of Mathematics, Ningxia University, Yinchuan 750021, PR China – sequence: 3 givenname: John R. surname: Schmitt fullname: Schmitt, John R. organization: Department of Mathematics, Middlebury College, Middlebury, VT, USA |
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| Cites_doi | 10.1007/s10114-005-0676-4 10.1016/S0095-8956(03)00044-3 10.21136/CPM.1955.108220 10.1007/BF02879940 10.1016/j.disc.2005.03.028 10.1016/S0012-365X(02)00765-3 10.1007/s00373-007-0737-9 10.1002/(SICI)1097-0118(199810)29:2<63::AID-JGT2>3.0.CO;2-A 10.1016/S0012-365X(99)00289-7 10.1360/02ys9076 10.1137/0110037 |
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| Issue | 24 |
| Keywords | Potentially F t , r , k -graphic sequence Generalized friendship graph Degree sequence Integer Graphics Algorithm theory Complete graph Subgraph Graph theory Combinatorics Potentially Ft,r,k-graphic sequence Graph algorithm |
| Language | English |
| License | http://www.elsevier.com/open-access/userlicense/1.0 https://www.elsevier.com/tdm/userlicense/1.0 https://www.elsevier.com/open-access/userlicense/1.0 CC BY 4.0 |
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| References | Gould, Jacobson, Lehel (b7) 1999; vol. I Havel (b9) 1955; 80 Lai (b10) 2007; 61 Li, Song, Luo (b13) 1998; 41 Yin, Li (b18) 2002; 45 J.R. Schmitt, On Potentially Yin, Li (b19) 2005; 301 Yin, Li (b17) 2003; 260 M.J. Ferrara, J.R. Schmitt, A sharp lower bound for potentially graphic sequences (submitted for publication) (under review) Ferrara (b4) 2007; 23 Hakimi (b8) 1962; 10 Erdős, Gallai (b2) 1960; 11 Li, Song (b12) 1998; 29 graphic Degree Sequences and Saturated Graphs, Ph.D. Dissertation, Emory University, May 2005 Erdős, Jacobson, Lehel (b3) 1991; vol. I Yin, Chen (b16) 2007; 72 Li, Song (b11) 2000; 212 Li, Yin (b14) 2006; 22 Chen, Gould, Pfender, Wei (b1) 2003; 89 Ferrara, Gould, Schmitt (b5) 2007; 85 Chen (10.1016/j.disc.2007.11.075_b1) 2003; 89 Gould (10.1016/j.disc.2007.11.075_b7) 1999; vol. I Ferrara (10.1016/j.disc.2007.11.075_b5) 2007; 85 Lai (10.1016/j.disc.2007.11.075_b10) 2007; 61 Li (10.1016/j.disc.2007.11.075_b13) 1998; 41 Yin (10.1016/j.disc.2007.11.075_b16) 2007; 72 Ferrara (10.1016/j.disc.2007.11.075_b4) 2007; 23 Yin (10.1016/j.disc.2007.11.075_b17) 2003; 260 Li (10.1016/j.disc.2007.11.075_b11) 2000; 212 Li (10.1016/j.disc.2007.11.075_b14) 2006; 22 Erdős (10.1016/j.disc.2007.11.075_b2) 1960; 11 Havel (10.1016/j.disc.2007.11.075_b9) 1955; 80 Erdős (10.1016/j.disc.2007.11.075_b3) 1991; vol. I Hakimi (10.1016/j.disc.2007.11.075_b8) 1962; 10 Li (10.1016/j.disc.2007.11.075_b12) 1998; 29 10.1016/j.disc.2007.11.075_b6 10.1016/j.disc.2007.11.075_b15 Yin (10.1016/j.disc.2007.11.075_b18) 2002; 45 Yin (10.1016/j.disc.2007.11.075_b19) 2005; 301 |
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Graph Theory doi: 10.1002/(SICI)1097-0118(199810)29:2<63::AID-JGT2>3.0.CO;2-A – volume: vol. I start-page: 451 year: 1999 ident: 10.1016/j.disc.2007.11.075_b7 article-title: Potentially G-graphical degree sequences – volume: 212 start-page: 223 year: 2000 ident: 10.1016/j.disc.2007.11.075_b11 article-title: An extremal problem on the potentially Pk-graphic sequence publication-title: Discrete Math. doi: 10.1016/S0012-365X(99)00289-7 – volume: 45 start-page: 694 year: 2002 ident: 10.1016/j.disc.2007.11.075_b18 article-title: The smallest degree sum that yields potentially Kr,r-graphic sequences publication-title: Science in China, Ser. A, doi: 10.1360/02ys9076 – volume: 10 start-page: 496 year: 1962 ident: 10.1016/j.disc.2007.11.075_b8 article-title: On the realizability of a set of integers as degrees of vertices of a graph publication-title: J. SIAM Appl. Math. doi: 10.1137/0110037 – volume: 11 start-page: 264 year: 1960 ident: 10.1016/j.disc.2007.11.075_b2 article-title: Graphs with given degrees of vertices publication-title: Math. Lapok – ident: 10.1016/j.disc.2007.11.075_b15 |
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| SubjectTerms | Algorithmics. Computability. Computer arithmetics Applied sciences Combinatorics Combinatorics. Ordered structures Computer science; control theory; systems Degree sequence Exact sciences and technology Generalized friendship graph Graph theory Information retrieval. Graph Mathematics Potentially [formula omitted]-graphic sequence Sciences and techniques of general use Theoretical computing |
| Title | Graphic sequences with a realization containing a generalized friendship graph |
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