An upper bound for a ramsey type problem for k-connected subgraphs
For any positive integer k , let r 2 ( k ) denote the smallest integer n such that every 2-edge-colored complete graph K n contains a monochromatic k -connected subgraph. Matula established the bound 4 ( k - 1 ) + 1 ≤ r 2 ( k ) < ( 3 + 11 / 3 ) ( k - 1 ) + 1 . It is known that r 2 ( k ) = 4 ( k -...
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| Published in: | Graphs and combinatorics Vol. 41; no. 6; p. 128 |
|---|---|
| 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: | For any positive integer
k
, let
r
2
(
k
)
denote the smallest integer
n
such that every 2-edge-colored complete graph
K
n
contains a monochromatic
k
-connected subgraph. Matula established the bound
4
(
k
-
1
)
+
1
≤
r
2
(
k
)
<
(
3
+
11
/
3
)
(
k
-
1
)
+
1
. It is known that
r
2
(
k
)
=
4
(
k
-
1
)
+
1
f
o
r
k
=
1
,
2
(by Bollobás and Gyárfás) and for
k
=
3
(by Liu, Morris, and Prince). We prove that for
k
≥
2
and
n
>
(
3
+
497
-
1
16
)
(
k
-
1
)
, every 2-edge-colored
K
n
contains a monochromatic
k
-connected subgraph with at least
2
(
k
-
1
)
vertices. This result improves the upper bound of
r
2
(
k
)
to
⌈
(
3
+
497
-
1
16
)
(
k
-
1
)
⌉
for all
k
≥
4
. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0911-0119 1435-5914 |
| DOI: | 10.1007/s00373-025-02993-8 |