2003 ICTM Contest Division AOrals Topic: Graph Theory

Micah Fogel

Illinois Math and Science Academy

The Source

Graphs And Their Uses

By Oystein Ore

(Rev. Robin J. Wilson)

Math. Assoc. of America

1990

Anneli Lax New Mathematical Library

Vol. 34

Topics

• Graph Theory– What is a graph?– Special properties

• Theory– Counting arguments– Trees and cycles– Special routes– Planar graphs

Topics (continued)

• Applications– Tournaments– Networks

• State Level Possibilities– Duals and geometric figures– Directed graphs– Matchings– Map coloring

Additional Resources

• Mailing list: fogel@imsa.edu

• Webstie: http://staff.imsa.edu/~fogel

• ICTM Website: www.viebach.net/ictm/admin/c2003r.htm

• Additional reading:– Bibliography– Discrete Math texts

Study Hints

• Do all exercises in text—many have solutions in back

• Keep an eye on the websites—especially for state topic list!

• Big area—don’t try to cover too much all at once– Go section-by-section

So what’s it all about…?

Here’s a graph!

Isomorphism

• Two graphs are isomorphic if you can pair the vertices of one with the vertices of the other so that the connections of a vertex are the same as for its paired vertex

Isomorphism

• To tell two graphs are not isomorphic, identify something different about them– Different numbers of edges emanating from

vertices– Different lengths of walks you can take on

them– Anything that can be done to one that can’t be

done to the other

Isomorphism

• To show graphs are isomorphic, you actually have to show the pairing(Example: the top two graphs are isomorphic, with pairing S Z and any matching of P, Q, R with W, X, Y. The bottom graph is not isomorphic to the other two.)

P

Q R

S

W X

Y

Z

Some Special Kinds of Graphs

• Complete graph has all possible edges

• Null graph has no edges

• Planar graph can be drawn on a plane with no crossing edges

• Interval graphs have vertices that are intervals of real numbers, edge between two intervals if they overlap

Counting Edges

• The degree of a vertex is the number of edges attached to it

• We can find the total number of edges in a graph by adding all the degrees and dividing by two (as each edge ends on two vertices)– This is often easier than counting all the edges

separately

• Graphs where each vertex has the same degree are called regular

Interval Graphs

(3,4)

(1,6)

(0,2) (5,7)Cannot be an interval graph

Routes in a Graph

• We can move from vertex to vertex along edges of a graph– If our route never reuses the same edge, it is

called a trail– If it never revisits the same vertex (except it may

end where it started) it is called a path– If the end and beginning are in the same place, it

is a cyclic trail, while a cyclic path is called a cycle

Connectivity

• All vertices that can be reached from a given starting point by travelling along some path are said to be connected to the starting vertex

• Being connected is an equivalence relation

• Equivalence classes are called components

Eulerian Graphs

• Can we find a trail running over each edge in the graph exactly once?– This problem is the place where graph theory was

born, in the famous Bridges of Königsberg

• It is easy to prove that if the number of vertices with odd degree is 2k, then we can find k trails that cover all the degrees

• In particular, if there are 0 or 2 odd vertices, there is a single Eulerian circuit

Eulerian Trails

Hamiltonian Cycles

• A similar question is: can we find a cycle that touches all vertices of a graph?– The Travelling Salesman Problem asks for such

a route, often also asking for the shortest such route if distances between vertices are given

– As opposed to the Eulerian circuit problem, this problem is considered “hard” to solve

Trees

• A tree is a connected graph with no cycle

• A forest is a graph each of whose components is a tree

• A forest with n vertices and k trees must therefore have n - k edges

Connector problem

• One example of trees is to find a minimum cost network to connect various nodes of the network– We can do this with the greedy algorithm—

always take the cheapest edge to connect the next city

– We can use this to approximate a solution to the Travelling Salesman Problem

Matching Problems

• A bipartite graph is a graph in which the vertices fall into two groups. No edge connects any two vertices in the same group

Matching Problems

• One common problem with bipartite graphs is to pair the vertices in the two subsets as efficiently as possible

• The Marriage Problem has a set of boys and a set of girls, and each person has a list of members of the opposite sex with whom they would be compatible. Is it possible to marry them off so that everyone is married to someone compatible?

Matching Problems

• Such a matching is possible if and only if the graph satisfies the diversity condition, that each subset of k boys is collectively compatible with at least k girls

Directed Graphs

• Often it is useful for the edges in a graph to have directionality associated with them– One way streets in a city road network– Who beat whom in a tournament graph

• Such graphs are called directed graphs or digraphs– Graphs that allow both directed and undirected

edges are called mixed graphs

Degrees of Vertices

• Now that edges are directed, vertices have two different kinds of degrees– In-degree is number of edges coming in to a vertex,

out-degree is the number of edges directed out of a vertex

– Now we have that the number of edges is the sum of the in-degrees, and also equals the sum of the out-degrees

– “Regular” changes its meaning so that all vertices must have the same in- and out-degrees

Applications of Digraphs

• The cycle structure of digraphs reveals whether they can be the graphs of family trees

• There is an efficient algorithm to find shortest routes (also known as minimal cost paths) in a network

Applications of Digraphs

• Many puzzles can be solved by brute force application of finding paths in digraphs– Puzzles such as pouring problems are amenable

to this technique– Games with complete information can be

solved by finding paths from the opening position to a winning position

Applications of Digraphs

• Sportswriter’s Paradox– In a complete directed graph, there is always a

path through every vertex– If there is no “outclassed set” we can always

find a Hamiltonian cycle in a complete digraph

Graphs and Relations

• Digraphs can be used to represent relations– We direct an edge from a to b whenever a is

related to b by our relation– The example below is the divisibility relation

on the numbers 1–7 (all directions point right). Primes are vertices with in-degree one)

1 2 3 4 5 6 7

Planar Graphs

• Many situations require that the graph that represents them must lie on a plane.– Printed circuit board networks– Streets where it is impractical to build

overpasses

• Is there a way to tell when a graph is planar?

Kuratowski’s Theorem

• Yes!

• An expansion of a graph is adding extra vertices in the middle of an edge. A contraction is deleting same

• A graph is planar if and only if no subgraph can be reduced to K5 or K3,3, the complete graph on five vertices or the complete bipartite graph on two sets of three vertices

Kuratowski’s Theorem

Euler’s Formula

• For any connected graph that lies in a plane, define a face to be a region of the plane “cordoned off” by edges of the graph

• Euler’s formula says that for any connected planar graph, f + v - e = 2

• Simply proved by induction on number of edges

Example: Water, Gas, Electicity

• For an example, we prove that the water, gas, and electricity puzzle has no planar solution

W G E

3 2 1

Example: Water, Gas, Electricity

• If it did, there would be six vertices and nine edges, thus five faces. But each face has four sides, requiring at least ten edges!

W G E

3 2 1

Example: Water, Gas, Electricity

• Note that the graph for the WGE problem would have to be a complete bipartite graph on two sets of three vertices, so can’t be planar by Kuratowski’s Theorem

• In fact, that’s how the “only if” part of Kuratowski’s theorem is proved!

Dual Graphs

• Given a planar graph G, we can make a new graph G* by putting a vertex of G* in each face of G, and connecting two vertices of G* if and only if the corresponding faces of G share an edge

• Text contains a nice proof that there can only be five perfect solids based on duals and an application of Euler’s formula

Coloring Problems

• Graphs give rise to a number of coloring problems, where we try to color the vertices, edges, or faces according to certain rules

• Most common are problems about coloring the faces of a graph (or, equivalently, the vertices of its dual)

Four Color Theorem

• A theorem, proved only in 1976, states that the faces of any planar graph can be colored so that no two neighboring faces have the same color using only four colors

• Text contains a straightforward proof that no more than five colors are needed

• No simple proof is known for four; the current best proof has several hundred special cases that need to be checked!