1. Testing k-colorability
Noga Alon and Michael Krivelevich
Presented by Igor Shinkar

2. Graph property testing
Let P be a graph property (bipartiteness, k- colorability, connectedness).
Given a graph G=(V,E) we wish to distinguish between the following:
G has the property P
G is ε-far from having P
The algorithm makes few queries to G
If G has the property P, then Pr[Alg accepts] > 2/3
If G is ε-far from P, then Pr[Alg rejects] > 2/3

3. Testing k-colorability
Given a dense graph G=(V,E) distinguish between
G is k-colorable.
G is ε-far from k-colorable, i.e. need to remove at least εn2 edges to make it k-colorable.
Strategy: Pick a random subgraph S and accept if the G[S] is k-colorable.

4. Testing k-colorability
The algorithm
Pick a random subset of vertices S, of size O(klg(k)/ε2).
Query all the edges in G[S].
Accept if and only if G[S] is k-colorable.
Clearly if G is k-colorable then all its subgraphs are also k-colorable.
So from now on we assume that G is ε-far from k-colorable.

5. G is ε-far from k-colorable
Intuition:
Let :S[k] be a k-coloring of S.
If u∈V\S has no feasible colors w.r.t. S and , we say that it is colorless w.r.t. S and .
If any coloring of u∈V\S restricts many vertices, we say that u is restricting w.r.t. S and .
We show that for any small set S, either the set of colorless vertices or the set of restricting vertices must be large.

6. Notation: restricting vertices
Given :S[k], denote by L(v) the feasible colors of v∈V\S w.r.t. (S,).
Let U be the set of colorless vertices w.r.t. (S,).
For v∈V\(SU) let (v,i) be the number of vertices that v restricts when colored with i.
(v,i) = |{uN(v)\(SU):iL(u)}|
By coloring v we shorten the lists of feasible colors for at least (v) = miniL(v) (v,i) vertices.

7. Illustration: restricting vertices
L(w)={1,2,4} w
L(x)={1,2,3} x
L(v)={1,2,3,4} V
(v,1)=|{w,x,z}| = 3 y
L(y)={2,3,4,5,7}
(v,2)=|{w,x,y}| = 3 z
L(z)={1,4,5,6}
(v,3)=|{x,y}| = 2
(v,4)=|{w,y,z}| = 3
By coloring v with 3 we restrict at most two vertices

8. Small U and   G is close to k-colorable
Claim 1 For every S and a k-coloring :S[k], the graph G is at most M edges far from k- colorable, where
M = (n-1)|SU| + ΣvV\(SU)(v)
Proof: Remove all edges touching SU.
For vSU remove (v) edges so that the coloring of v will restrict none of it neighbors.

9. G is ε-far from k-colorable
Corollary 1
If G is ε-far from k-colorable, then
ΣvV\(SU)(v) > εn2 – n(|S|+|U|)
Corollary 2
Let W={v∈V\(SU) : (v) > εn/2} be the set of restricting vertices. Then
|U|+|W|+|S| > εn/2

10. G is ε-far from k-colorable
Corollary 2
Let W={v∈V\(SU) : (v) > εn/2} be the set of restricting vertices. Then
|UW|> εn/2 -|S|
That means that either there is a large set of colorless vertices or there is a large set of restricting vertices.

11. Defining the auxiliary k-tree
The nodes of T are labeled with either v∈V or #. Nodes labeled with # are always leaves.
The edges are colored with [k].
For a node t in T, let S(t) be the set of labels on the path from the root to t.
Let (t):S(t)[k] be the coloring of S(t) defined according to the color of the edge along the path.
The child of t is labeled # if the corresponding color is infeasible for t w.r.t. (S(t),(t)).

12. Defining the auxiliary k-treeb c # d # #
 ={ac} # #
={abd}
={abd}
Red is infeasible for d

13. Defining the auxiliary k-tree
Start with a single unlabeled node.
At the i’th step:
Pick a random vertex vi=d.
For each unlabeled node t consider (S(t),(t)).
If d is in UW, call round i successful and set the label of t to d and update her children. a
dUW b # c d # # # d
d∈U
d∈W # # # #

14. Bounding the depth of the tree
Claim 3 The depth of T is bounded by 2k/ε. In particular all leaves on level 2k/ε are #.
Proof: Fix a node t in T and let t1,t2,…tr=t be the path from the root to t.
Each node ti is a restricting vertex w.r.t. (S(ti),(ti)). Thus removing at least εn/2 colors from the list of all feasible colors.
Recall that the list of all colors is at most kn.

15. Two simple observations
Observation 1: If a leaf t is labeled #, then the coloring (t):S(t)[k] is not proper.
Observation 2: Suppose after s steps the sampled vertices of G are v1,v2,…,vs. If all leaves of T are labeled #, then the subgraph induced by v1,v2,…,vs is not k-colorable.

16. Proof of the theorem
Claim 4 If G is ε-far from k-colorable, then w.h.p. after s = O(klg(k)/ε2) steps all leaves are labeled #, i.e. G[S] is not k-colorable.
Proof: The number of leaves in T is≥ k2k/ε.
At each step the probability of a node to be successful is >ε/2
For every path p=t1…tr (r<2k/ε) Pr[label(tr) ≠ #] < Pr[Bin(s, ε/2) < 2k/ε]

17. Lower bound
Theorem: For every k≥3 and ε>0 small enough there are infinitely many graphs G such that
1) G is ε-far from being k-colorable.
2) Every Ω(1/ε) vertices induce a k-colorable subgraph.

18. Lower bound: Key lemma
Lemma: For every k≥3 and sufficiently large m, there exists a graph H=Hk,m such that
1) Every subset on at most m/lg(k) vertices of H is k-colorable.
2) Every subset U of size |U|>m/2k span at least Ω(lg(k)|U|) edges.
3) H is Ω(lg(k)m) edges far from k-colorable.

19. So what did we show?
If G is ε-far from k-colorable, then w.h.p. a subgarph of size O(klg(k)/ε2) is not k-colorable.
There exist graphs ε-far from k-colorable, such that any subgraph of size Ω(1/ε) is k-colorable.

20. FIN