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research
Vertex covering with monochromatic pieces of few colours
Authors
Marlo Eugster
Frank Mousset
Publication date
12 August 2018
Publisher
Doi
Cite
View
on
arXiv
Abstract
In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every
2
2
2
-colouring of the edges of
K
n
K_n
K
n
β
, there is a vertex cover by
2
n
2\sqrt{n}
2
n
β
monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers
r
,
s
r,s
r
,
s
, what is the smallest number
pc
r
,
s
(
K
n
)
\text{pc}_{r,s}(K_n)
pc
r
,
s
β
(
K
n
β
)
such that in every colouring of the edges of
K
n
K_n
K
n
β
with
r
r
r
colours, there exists a vertex cover of
K
n
K_n
K
n
β
by
pc
r
,
s
(
K
n
)
\text{pc}_{r,s}(K_n)
pc
r
,
s
β
(
K
n
β
)
monochromatic paths using altogether at most
s
s
s
different colours? For fixed integers
r
>
s
r>s
r
>
s
and as
n
β
β
n\to\infty
n
β
β
, we prove that
pc
r
,
s
(
K
n
)
=
Ξ
(
n
1
/
Ο
)
\text{pc}_{r,s}(K_n) = \Theta(n^{1/\chi})
pc
r
,
s
β
(
K
n
β
)
=
Ξ
(
n
1/
Ο
)
, where
Ο
=
max
β‘
{
1
,
2
+
2
s
β
r
}
\chi=\max{\{1,2+2s-r\}}
Ο
=
max
{
1
,
2
+
2
s
β
r
}
is the chromatic number of the Kneser gr aph
KG
(
r
,
r
β
s
)
\text{KG}(r,r-s)
KG
(
r
,
r
β
s
)
. More generally, if one replaces
K
n
K_n
K
n
β
by an arbitrary
n
n
n
-vertex graph with fixed independence number
Ξ±
\alpha
Ξ±
, then we have
pc
r
,
s
(
G
)
=
O
(
n
1
/
Ο
)
\text{pc}_{r,s}(G) = O(n^{1/\chi})
pc
r
,
s
β
(
G
)
=
O
(
n
1/
Ο
)
, where this time around
Ο
\chi
Ο
is the chromatic number of the Kneser hypergraph
KG
(
Ξ±
+
1
)
(
r
,
r
β
s
)
\text{KG}^{(\alpha+1)}(r,r-s)
KG
(
Ξ±
+
1
)
(
r
,
r
β
s
)
. This result is tight in the sense that there exist graphs with independence number
Ξ±
\alpha
Ξ±
for which
pc
r
,
s
(
G
)
=
Ξ©
(
n
1
/
Ο
)
\text{pc}_{r,s}(G) = \Omega(n^{1/\chi})
pc
r
,
s
β
(
G
)
=
Ξ©
(
n
1/
Ο
)
. This is in sharp contrast to the case
r
=
s
r=s
r
=
s
, where it follows from a result of S\'ark\"ozy (2012) that
pc
r
,
r
(
G
)
\text{pc}_{r,r}(G)
pc
r
,
r
β
(
G
)
depends only on
r
r
r
and
Ξ±
\alpha
Ξ±
, but not on the number of vertices. We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic
d
d
d
-regular graphs
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Last time updated on 19/04/2020