Problem 4.9.15. Let G be a connected graph and for each edge e let w(e) be some real number. Let T be a spanning tree of G such that for every edge f ∈ E(G) − E(T ), w(f ) ≥ w(e) for every edge e of the fundamental cycle of f . Show that T is a min-cost tree.
Problem 4.9.16. Suppose graph G with n vertices contains at least 4n3 edges. Then G contains two intersecting cycles.
Problem 4.9.17. Prove that a graph with at most two odd cycles has chromatic number of at most 3.
Problem 4.9.18. Let G be a graph where every two odd circles have at least a vertex in common. Prove that G is 5-colorable.
Problem 4.9.19. Let Qn denote the graph whose vertex set is {0, 1}n (i.e. there are exactly 2n, each labeled with a distinct n-bit string), and with an edge between vertices x and y if and only if x and y differ in exactly one coordinate. This is called the n-dimensional hypercube.
a) If n is even, prove that Qn has an Eulerian circuit.
b) In general, prove that Qn has a Hamiltonian path.
c) Does (b) resemble anything you have seen before?
§4.10 Problems: Planar Graphs
These are some problems to practice the material above and do not represent homework unless explicitly mentioned otherwise. Give them a try! Some of them will be discussed by your TA during the upcoming discussion sessions from 4 to 5 PM on Tuesdays and Thursdays.
Problem 4.10.20. Prove that any simple, connected planar graph contains a vertex of degree at most 5.
Problem 4.10.21. Prove that any simple, connected planar graph is 6-colorable.
Problem 4.10.22. Improve the argument from the previous exercise and show that any simple, connected planar graph is 5-colorable. (Hint: You can use Kuratowski’s theorem).
Problem 4.10.23. A graph is a minor of a graph G if H can be obtained from a subgraph of G by a sequence of edge contractions. Prove that K5 and K3,3 are both minors of the Petersen graph.
Problem 4.10.24. Prove that a simple graph G is a tree if and only if G has no loop as a minor.
Problem 4.10.25. Prove that a graph G is planar if and only if K5 and K3,3 are not minors of G. Note that this is weaker than Kuratowski’s theorem, thus prove it from first principles.
4.10 Problems: Planar Graphs
Problem 4.10.26. Graph G is outerplanar if it can be drawn in the plan so that every vertex is incident with the infinite region. Show that a graph G is outerplanar if and only if G has no K4 or K2,3 minor.
Problem 4.10.27. Prove that if G is planar then G is a minor of a large enough grid graph. Formally, an r×r grid graph is a graph H(V, E), where V = {1, . . . , r}×{1, . . . , r}
and two pairs (i, j) and (i0, j0) are connected by an edge if and only if |i − i0| + |j − j0| = 1.
More intuitively, they’re just the graphs associated with the geometric square lattice grids.)
5 Extremal graph / Ramsey theory
Complete disorder is impossible.
—Theodore Motzkin One of the first results in extremal graph theory (before it was even called so) is Mantel’s theorem, proved in 1907, which states that any graph on n vertices with no triangle contains at most n2/4 edges. This is clearly the best possible, as one may partition the set of n vertices into two sets of size n/2 (floor and ceilling if n is not even) and form the complete bipartite graph between them.
Theorem 5.0.1 (Mantel’s theorem)
If a graph G on n vertices contains no triangle then it contains at most n42 edges.
Proof 1. Suppose that G has m edges. Let x and y be two vertices in G which are joined by an edge. If d(v) is the degree of a vertex v, we see that d(x) + d(y) ≤ n. This is because every vertex in the graph G is adjacent to at most one of x and y. Note now that
X
x
d2(x) = X
xy∈E
(d(x) + d(y)) ≤ mn.
On the other hand, sinceP
xd(x) = 2m, the Cauchy-Schwarz inequality implies that X
x
d2(x) ≥ (P
xd(x))2
n ≥ 4m2
n . Therefore
4m2
n ≤ mn, and the result follows.
There’s also a nice short proof with induction.
Proof 2. We proceed by induction on n. For n = 1 and n = 2 the result is trivial, so assume we know it to be true for n − 1 and let G be a graph on n vertices. Let x and y be two adjacent vertices in G. As above, we know that d(x) + d(y) ≤ n. The complement H of x and y has n − 2 vertices and sicne it contains no triangles, it must have by induction at most (n − 2)2/4 edges. Therefore, the total number of edges in G is at most
e(H) + d(x) + d(y) − 1 ≤ (n − 2)2
4 + n − 1 = n2 4 ,
where the −1 comes from the fact that we count the edge between x and y twice.
The latter proof generalizes and can be adapted to show the more general result by Turan that lies at the heart of extremal graph theory.
5.1 Turan’s theorem will therefore assume that it is true for all values less than n and prove it for n. Let G be a graph on n vertices which contains no Kr+1 and has the maximum possible number of edges. Then G contains copies of Kr. Otherwise, we could add edges to G, contradicting maximality.
Let A be a clique of size r and let B be its complement. Since B has size n − r and contains no Kr+1, there are at most 1 − 1r(n−r)2
2 edges in B. Moreover, since every vertex in B can have at most r − 1 neighbors in A, the number of edges between A And B is at most (r − 1)(n − r). Summing, we see that
In general for any fixed graph H, we can ask ourselves the question: how many edges can a given graph G have if G does not contain any copy of H as a subgraph? There are a bunch of interesting theorems for many specials graphs H. Below are some examples:
Example 5.1.2 a) Prove that every graph on n ≥ 4 vertices and m > n+n
√4n−3 4
edges has at least one 4-cycle.
b) If a graph G has n vertices and does not contain a K2,3 as a subgraph, then G has no more than n+n
√8n−7
4 edges.
Proof. a) We count triples (c, a, b) where a, b, c are vertices such that c is connected to both a and b. In other words, we double count cherries. For a fixed vertex c we have d(c)2− d(c) possibilities for the pair (a, b), so there are at leastP
c
(d(c)2− d(c)) triples.
By the Cauchy-Schwarz inequality, if m is the number of edges of the graph, then X
c
d(c)2− d(c) ≥ 4m2 n − 2m.
If there are no 4-cycles, then for fixed a, b there is at most one vertex c that figures in a triple (a, b, c), so there are at most n(n − 1) triples and so 4mn2 − 2m ≤ n2− n, which implies m ≤ n+n
√4n−3
4 , a contradiction.
b) Count ordered triples (A, B, C) where vertex C is connected to both vertices A and B. For each ordered pair (A, B) there are at most 2 triples of the form (A, B, C) by hypothesis. So in total we have at most 2n(n − 1) triples. On the other hand, if d(A) is the degree of A, then for each C we have d(C)(d(C) − 1) triples of the form (A, B, C).
Hence
X
C
d(C)(d(C) − 1) ≤ 2n(n − 1).
5.1 Turan’s theorem
The rest follows by Cauchy-Schwarz and the fact that P
Cd(C) is twice the number of edges. all connected to v (”generalized cherries”). For each v = i, there are precisely dsi sets {v1, ..., vs} sharing a pair with v. Thus we findPn
Here we used Jensen’s inequality. This yields the rough estimate t · ns≥ n 2q
n − s + 1
s
, from which the result follows immediately.
Exercise 5.1.4. Can one adapt this argument to generalize part (b) from Example 8.8?
Time for two side puzzles.
Let P be a set of n points in the plane such that no three points are collinear. Prove that P determines at least cn distinct distances for some absolute constant c > 0 (independent of n).
5.1 Turan’s theorem
What is the connection between them and Kovari-Sos-Turan?
The answer is (hopefully) going to surprise you: they have the same proof, double counting cherries! I will only address the second example, since it is in fact a theorem from an important topic in combinatorial geometry. We will count the number T of triangles ABC, where A, B, C are distinct points in S such that AB = AC. In other words, the cherries here are nothing but isosceles triangles with vertices in P . First, for any pair {B, C} of elements of P there are at most 2 isosceles triangles ABC with A ∈ P , since the points A such that AB = AC all lie on a line, and no three points of P
Taking the sum over A ∈ P and using the first paragraph, we obtain X
On the other hand, by Cauchy-Schwarz we have that
k · Returning to general graphs H, there is in fact a very strong general theorem that is worth mentioning.
Theorem 5.1.7 (Erd¨os-Stone-Simonovits)
For any graph H, if r denotes the chromatic number of H (namely χ(H) = r), we have that