Minors in expanding graphs
M. Krivelevich and B. Sudakov
Speakers: A. Treglown and F. Knox
Two central topics in modern graph theory are minors and expanding graphs.
Much research has gone on in both areas although these subjects are normally
somewhat disjoint.
Let G and H be graphs and let V (H) = {u1 , . . . , uk }. We say that
H is a minor of G if V (G) contains disjoint subsets A1 , . . . , Ak such that
each of the graphs G[Ai ] are connected and there is an edge between Ai and
Aj in G whenever the corresponding vertices ui and uj are adjacent in H.
Equivalently, we can consider a minor H of G as a graph obtained from G
through edge contractions and deletions: We can obtain H from G by firstly
deleting all edges in G except those in the graphs G[Ai ] and those edges
between sets Ai and Aj that correspond to adjacent vertices in H. We then
contract each of our components Ai into a single vertex. (Given an edge uv
of a graph G contracting uv results in replacing u and v by a single vertex
w whose neighbourhood is the union of the neighbourhoods of u and v.)
There are several notions of expansion in use, many of which capture
somewhat different expansion properties. Informally, a graph G is an expander if certain subsets X ⊆ V (G) have relatively many neighbours outside
of X. The following definition describes an expansion property in small sets:
Let t, α > 0. A graph G is (t, α)-expanding if given any set X ⊆ V (G) where
|X| ≤ α|V |/t we have that the external neighbourhood N (X)\X has size at
least t|X|. The following result states that such expanding graphs contain
minors of “large” average degree.
Theorem 1 Let 0 < α < 1 be a constant. Let G be a (t, α)-expanding graph
of order n, and let t ≥ 10. Then G contains a minor with average degree at
least
√
3 nt log t
,
cα √
log n
where c > 0 is an absolute constant independent of α.
Using results of Kostochka and Thomason one can obtain the following corollary.
Corollary 2 Let 0 < α < 1 be a constant. Let G be a (t, α)-expanding graph
of order n, and let t ≥ 10. Then G contains a clique minor of size
√
3 nt log t
cα √
,
log n
where c > 0 is an absolute constant independent of α.
A notion closely linked to expanding graphs is that of pseudo-random graphs.
(As the name suggests, very roughly, these are the graphs which resemble random graphs.) Thomason formulated the following notion of psuedo-random
graphs: Let p, β > 0. A graph G is (p, β)-jumbled if for every X ⊆ V (G) we
have that |eG (X) − p|X|2 /2| ≤ β|X|. Note that (p, β)-jumbled graphs have
density close to p and have edge distribution similar to the random graph
Gn,p .
Theorem 3 Let G be a (p, β)-jumbled graph of order n such that β = o(np).
√
Then G contains a minor with average degree cn p, where c > 0 is an
absolute constant.
The following two results give analogues of Theorems 1 and 3 for graphs
which do not contain either a Ks,s0 or a C2k as a subgraph.
Theorem 4 Let 2 ≤ s ≤ s0 be integers. Let G be a graph with average degree
r which does not contain Ks,s0 as a subgraph. Then G contains a minor with
1
average degree cr1+ 2(s−1) , where c = c(s, s0 ) > 0 is a constant.
Theorem 5 Let k ≥ 2 and let G be a graph with average degree r which
does not contain C2k as a subgraph. Then G contains a minor with average
k+1
degree cr 2 , where c = c(k) > 0 is a constant.
In our talks we will give an outline of the proofs of some of the results
discussed above.