Author:
Kim Jaehoon,Lee Joonkyung,Liu Hong,Tran Tuan
Abstract
AbstractWe prove that every properly edge-colored n-vertex graph with average degree at least $$32(\log 5n)^2$$
32
(
log
5
n
)
2
contains a rainbow cycle, improving upon the $$(\log n)^{2+o(1)}$$
(
log
n
)
2
+
o
(
1
)
bound due to Tomon. We also prove that every properly edge-colored n-vertex graph with at least $$10^5 k^3 n^{1+1/k}$$
10
5
k
3
n
1
+
1
/
k
edges contains a rainbow 2k-cycle, which improves the previous bound $$2^{ck^2}n^{1+1/k}$$
2
c
k
2
n
1
+
1
/
k
obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős–Simonovits supersaturation theorem for even cycles, which may be of independent interest.
Funder
Korea Advanced Institute of Science and Technology
Publisher
Springer Science and Business Media LLC
Reference9 articles.
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