Author:
Janzer Oliver,Sudakov Benny,Tomon István
Abstract
AbstractIn this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number $$s>2$$
s
>
2
, we prove that every graph on n vertices with average degree $$d\ge s$$
d
≥
s
contains a subgraph of average degree at least s on at most $$nd^{-\frac{s}{s-2}}(\log d)^{O_s(1)}$$
n
d
-
s
s
-
2
(
log
d
)
O
s
(
1
)
vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with n vertices and average degree at least $$n^{1-\frac{2}{s}+\varepsilon }$$
n
1
-
2
s
+
ε
contains a subgraph of average degree at least s on $$O_{\varepsilon ,s}(1)$$
O
ε
,
s
(
1
)
vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraëte.
Publisher
Springer Science and Business Media LLC
Reference31 articles.
1. Alon, N., Hoory, S., Linial, N.: The Moore bound for irregular graphs. Graphs Comb. 18(1), 53–57 (2002)
2. Arora, S., Lovász, L., Newman, I., Rabani, Y., Rabinovich, Y., Vempala, S.: Local versus global properties of metric spaces. SIAM J. Comput. 41(1), 250–271 (2012)
3. Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities – an $$O(n^{1/4})$$ approximation for densest $$k$$-subgraph, Proceedings of STOC (2010)
4. Bhaskara, A., Charikar, M., Guruswami, V., Vijayaraghavan, A., Zhou, Y.: Polynomial integrality gaps for strong SDP relaxations of Densest $$k$$-subgraph, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA12)
5. Bollobás, B.: Extremal graph theory, Dover Books on Mathematics Series. Dover Publications, Mineola (2004)
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