Hamiltonian Cycles Avoiding a Spanning Forest or a Spanning Cycle Subgraph in Tournaments
Bang-Jensen, Gutin and Yeo [Combin. Probab. Comput. 6(3) (1997) 255–261] investigated hamiltonian cycles avoiding the union of disjoint cliques in tournaments: for a k -strong tournament T = ( V , A ) on n vertices, a partition X 1 , X 2 , … , X p of V with | X 1 | ≤ | X 2 | ≤ ⋯ ≤ | X p | , and a di...
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| Published in: | Graphs and combinatorics Vol. 41; no. 6; p. 129 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Tokyo
Springer Japan
01.12.2025
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0911-0119, 1435-5914 |
| Online Access: | Get full text |
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| Summary: | Bang-Jensen, Gutin and Yeo [Combin. Probab. Comput. 6(3) (1997) 255–261] investigated hamiltonian cycles avoiding the union of disjoint cliques in tournaments: for a
k
-strong tournament
T
=
(
V
,
A
)
on
n
vertices, a partition
X
1
,
X
2
,
…
,
X
p
of
V
with
|
X
1
|
≤
|
X
2
|
≤
⋯
≤
|
X
p
|
, and a digraph
D
obtained from
T
by deleting all arcs which have both head and tail in the same
X
i
(i.e.,
D
=
T
-
∪
i
=
1
p
A
(
T
[
X
i
]
)
), if
|
X
p
|
≤
n
/
2
and
k
≥
|
X
p
|
+
∑
i
=
1
p
-
1
|
X
i
|
/
2
, then
D
is hamiltonian. They showed the bound on
k
is the best possible and raised the problem: which sets
B
of edges of the complete graph
K
n
have the property that every
k
-strong orientation of
K
n
induces a hamiltonian digraph on
K
n
-
B
? The above result provides the sharp bound of
k
when
B
is the union of cliques. In particular, they asked what are sharp bounds for
k
when
B
is a spanning forest (or a spanning cycle subgraph) of
K
n
, consisting of
p
disjoint paths, or
p
disjoint stars (or
p
disjoint cycles) containing
r
1
,
…
,
r
p
vertices, respectively. In this paper, we give the bounds for
k
on the above problems and prove these bounds are almost best possible, when each component of the spanning forest (or spanning cycle subgraph) has at most 3 vertices. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0911-0119 1435-5914 |
| DOI: | 10.1007/s00373-025-02991-w |