Cycle Lengths of Hamiltonian Pℓ-free Graphs
For an integer ℓ at least three, we prove that every Hamiltonian P ℓ -free graph G on n > ℓ vertices has cycles of at least 2 ℓ n - 1 different lengths. For small values of ℓ , we can improve the bound as follows. If 4 ≤ ℓ ≤ 7 , then G has cycles of at least 1 2 n - 1 different lengths, and if ℓ...
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| Published in | Graphs and combinatorics Vol. 31; no. 6; pp. 2335 - 2345 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Tokyo
Springer Japan
05.11.2014
|
| Subjects | |
| Online Access | Get full text |
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| Abstract | For an integer
ℓ
at least three, we prove that every Hamiltonian
P
ℓ
-free graph
G
on
n
>
ℓ
vertices has cycles of at least
2
ℓ
n
-
1
different lengths. For small values of
ℓ
, we can improve the bound as follows. If
4
≤
ℓ
≤
7
, then
G
has cycles of at least
1
2
n
-
1
different lengths, and if
ℓ
is
4
or
5
and
n
is odd, then
G
has cycles of at least
n
-
ℓ
+
2
different lengths. |
|---|---|
| AbstractList | For an integer
ℓ
at least three, we prove that every Hamiltonian
P
ℓ
-free graph
G
on
n
>
ℓ
vertices has cycles of at least
2
ℓ
n
-
1
different lengths. For small values of
ℓ
, we can improve the bound as follows. If
4
≤
ℓ
≤
7
, then
G
has cycles of at least
1
2
n
-
1
different lengths, and if
ℓ
is
4
or
5
and
n
is odd, then
G
has cycles of at least
n
-
ℓ
+
2
different lengths. |
| Author | Rautenbach, Dieter Meierling, Dirk |
| Author_xml | – sequence: 1 givenname: Dirk surname: Meierling fullname: Meierling, Dirk organization: Institut für Optimierung und Operations Research, Ulm University – sequence: 2 givenname: Dieter surname: Rautenbach fullname: Rautenbach, Dieter email: dieter.rautenbach@uni-ulm.de organization: Institut für Optimierung und Operations Research, Ulm University |
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| Cites_doi | 10.2307/2308928 10.1016/0095-8956(71)90016-5 10.1016/0095-8956(88)90058-5 10.1016/0095-8956(90)90133-K 10.1007/s00373-012-1156-0 10.1016/0095-8956(72)90020-2 10.1016/S0012-365X(02)00817-8 10.1016/j.jctb.2012.04.002 10.1016/0095-8956(84)90054-6 |
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| DOI | 10.1007/s00373-014-1494-1 |
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| Keywords | Cycle length Cycle spectrum 05C45 Hamiltonian cycle 05C38 |
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| References | Müttel, Rautenbach, Regen, Sasse (CR8) 2013; 29 Marczyk, Woźniak (CR7) 2003; 266 Chvátal (CR4) 1972; 12 Ore (CR9) 1960; 67 CR3 Bauer, Schmeichel (CR1) 1990; 48 Fan (CR5) 1984; 37 Schmeichel, Hakimi (CR10) 1988; 45 Bondy (CR2) 1971; 11 Milans, Pfender, Rautenbach, Regen, West (CR6) 2012; 102 |
| References_xml | – volume: 67 start-page: 55 year: 1960 ident: CR9 article-title: Note on Hamilton circuits publication-title: Am. Math. Monthly doi: 10.2307/2308928 contributor: fullname: Ore – volume: 11 start-page: 80 year: 1971 end-page: 84 ident: CR2 article-title: Pancyclic graphs publication-title: J. Combin. Theory (B) doi: 10.1016/0095-8956(71)90016-5 contributor: fullname: Bondy – volume: 45 start-page: 99 year: 1988 end-page: 107 ident: CR10 article-title: A cycle structure theorem for Hamiltonian graphs publication-title: J. Combin. Theory (B) doi: 10.1016/0095-8956(88)90058-5 contributor: fullname: Hakimi – volume: 48 start-page: 111 year: 1990 end-page: 116 ident: CR1 article-title: Hamiltonian degree conditions which imply a graph is pancyclic publication-title: J. Combin. Theory (B) doi: 10.1016/0095-8956(90)90133-K contributor: fullname: Schmeichel – volume: 29 start-page: 1067 year: 2013 end-page: 1076 ident: CR8 article-title: On the cycle spectrum of cubic Hamiltonian graphs publication-title: Graphs Comb. doi: 10.1007/s00373-012-1156-0 contributor: fullname: Sasse – ident: CR3 – volume: 12 start-page: 163 year: 1972 end-page: 168 ident: CR4 article-title: On Hamiltons ideals publication-title: J. Combin. Theory (B) doi: 10.1016/0095-8956(72)90020-2 contributor: fullname: Chvátal – volume: 266 start-page: 321 year: 2003 end-page: 326 ident: CR7 article-title: Cycles in hamiltonian graphs of prescribed maximum degree publication-title: Disc. Math. doi: 10.1016/S0012-365X(02)00817-8 contributor: fullname: Woźniak – volume: 102 start-page: 869 year: 2012 end-page: 874 ident: CR6 article-title: Cycle spectra of Hamiltonian graphs publication-title: J. Combin. Theory (B) doi: 10.1016/j.jctb.2012.04.002 contributor: fullname: West – volume: 37 start-page: 221 year: 1984 end-page: 227 ident: CR5 article-title: New sufficient conditions for cycles in graphs publication-title: J. Combin. Theory (B) doi: 10.1016/0095-8956(84)90054-6 contributor: fullname: Fan |
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| Snippet | For an integer
ℓ
at least three, we prove that every Hamiltonian
P
ℓ
-free graph
G
on
n
>
ℓ
vertices has cycles of at least
2
ℓ
n
-
1
different lengths. For... |
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| SubjectTerms | Combinatorics Engineering Design Mathematics Mathematics and Statistics Original Paper |
| Title | Cycle Lengths of Hamiltonian Pℓ-free Graphs |
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