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More on Rainbow Cliques in Edge-Colored Graphs
Authors
Xiao-Chuan Liu
Danni Peng
Xu Yang
Publication date
14 August 2023
Publisher
View
on
arXiv
Abstract
In an edge-colored graph
G
G
G
, a rainbow clique
K
k
K_k
K
k
β
is a
k
k
k
-complete subgraph in which all the edges have distinct colors. Let
e
(
G
)
e(G)
e
(
G
)
and
c
(
G
)
c(G)
c
(
G
)
be the number of edges and colors in
G
G
G
, respectively. In this paper, we show that for any
Ξ΅
>
0
\varepsilon>0
Ξ΅
>
0
, if
e
(
G
)
+
c
(
G
)
β₯
(
1
+
k
β
3
k
β
2
+
2
Ξ΅
)
(
n
2
)
e(G)+c(G) \geq (1+\frac{k-3}{k-2}+2\varepsilon) {n\choose 2}
e
(
G
)
+
c
(
G
)
β₯
(
1
+
k
β
2
k
β
3
β
+
2
Ξ΅
)
(
2
n
β
)
and
k
β₯
3
k\geq 3
k
β₯
3
, then for sufficiently large
n
n
n
, the number of rainbow cliques
K
k
K_k
K
k
β
in
G
G
G
is
Ξ©
(
n
k
)
\Omega(n^k)
Ξ©
(
n
k
)
. We also characterize the extremal graphs
G
G
G
without a rainbow clique
K
k
K_k
K
k
β
, for
k
=
4
,
5
k=4,5
k
=
4
,
5
, when
e
(
G
)
+
c
(
G
)
e(G)+c(G)
e
(
G
)
+
c
(
G
)
is maximum. Our results not only address existing questions but also complete the findings of Ehard and Mohr (Ehard and Mohr, Rainbow triangles and cliques in edge-colored graphs. {\it European Journal of Combinatorics, 84:103037,2020}).Comment: 16page
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oai:arXiv.org:2308.07405
Last time updated on 18/08/2023