AMS 301.01
Finite Mathematical Structures
Chapter 1: Elements of Graph Theory
Figures taken from Tucker, Applied Combinatorics and Harris, Hirst & Mossinghoff, Combinatorics and
Graph Theory.
Section 1.1: Graph Models
Def: A graph G = (V, E) consists of a finite set V of vertices and a set E of edges joining different pairs
of distinct vertices.
Figure 1: A visual representation of the graph G with vertices V = {a, b, c, d, e, f, g, h} and edges E =
{(a, d), (a, e), (b, c), (b, e), (b, g), (c, f ), (d, f ), (d, g), (g, h)}.
Def: Vertices a and b are adjacent when there is an edge (a, b).
Def: A directed graph D = (V, A) consists of a finite set of vertices V and a set of ordered pairs of
directed edges.
In a directed graph, one edge is allowed in each direction between a pair of vertices. A directed edge from
b to c is denoted by (b~, c).
Figure 2: A directed graph G. Note that G has edges including (a~, c) and (c~, a).
Def: A multigraph G = (V, A) consists of a finite set of vertices V and a set of pairs of edges joining pairs
of vertices, allowing loops (an edge connecting a vertex to itself) and multiple edges (copies of an edge
connecting the same pair of vertices).
Figure 3: A multigraph.
AMS 301.01
Chapter 1: Elements of Graph Theory
Figure 4
Def: A path P is a sequence of distinct vertices, written P = x1 −x2 −· · ·−xn , with each pair of consecutive
vertices P joined by an edge.
Examples of paths in the graph in Figure 4:
Def: A circuit is a path that begins and ends at the same vertex.
Examples of paths in the graph in Figure 4:
Def: A graph is connected if there exists a path between every pair of vertices.
A graph that is not connected is also called disconnected.
Figure 5: A disconnected graph on 7 vertices.
Def: A bipartite graph is a graph whose vertices can be partitioned into two sets (X and Y ) and each
edge connects vertices in different sets.
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Chapter 1: Elements of Graph Theory
Figure 6: Two bipartite graphs and one non-bipartite graph.
Def: A complete graph Kn is a graph on n vertices in which there exists an edge between every pair of
vertices.
Figure 7: Examples of complete graphs.
Example: Draw K1 and K4 .
Def: A complete bipartite graph Kn,m is a bipartite graph whose edge set is of the form E = {xy|x ∈
X, y ∈ Y } (that is, every possible connection of a vertex of X with a vertex of Y is present in the
graph.)
Figure 8: Examples of complete bipartite graphs.
Example: Draw K4,1 .
Def: The degree of a vertex is the number of edges incident to that vertex.
For a multigraph, a loop counts twice toward the degree of that vertex.
For a directed graphs, we can define in-degree and out-degree.
Def: A set of vertices with no edges existing between any two is called an independent set of vertices (i.e.
a set of vertices such that for every pair of vertices v, w there does not exist an edge (v, w) ∈ E.)
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Chapter 1: Elements of Graph Theory
Note: If you are ever asked to model something with a graph, be sure to state (1) what the vertices
represents (e.g. people) and (2) what the edges represent (e.g. there exists an edge between vertices a
and b if and only if persons a and b are friends).
Example: Suppose interstate highways join the six towns A, B, C, D, E, F as follows: I-77 goes from B
through A to E; I-82 goes from C through D, then through B to F ; I-85 goes from D through A to
F ; I-90 goes from C through E to F ; and I-91 goes from D to E.
a. Draw a graph of the network with vertices for towns and edges for segments of interstates linking
neighboring towns.
b. What is the minimum number of edges whose removal prevents travel between some pair of towns
(i.e. disconnects the graph)?
c. Is it possible to take a trip starting from town C that goes to every town without using any
interstate highway for more than one edge (the trip does not need to return to C)?
Figure 9
Example: Find the minimal connecting set of edges in Figure 9 (i.e. the smallest set of edges needed to
link together the 11 vertices).
Def: In undirected graphs, a tree is a connected graph with no circuits.
A minimal connecting set will always be a tree.
If a graph G has n vertices, then a minimal connecting set for G (if an exists) always has n − 1 edges.
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Chapter 1: Elements of Graph Theory
Def: A set C of vertices in a graph G with the property that every edge of G is incident to at least one
vertex in C is called an edge cover.
Section 1.2: Isomorphism
Def: Two graphs G and G0 are isomorphic if there exists a one-to-one correspondence between the vertices
in G and the vertices in G0 such that a pair of vertices are adjacent in G if and only if the corresponding
pair of vertices are adjacent in G0 .
Such a one-to-one correspondence of vertices that preserves adjacency is called an isomorphism.
To be isomorphic, two graphs must have the same number of vertices and the same number of edges.
Figure 10
Example: Find an isomorphism between the graphs in Figure 10.
Degrees are preserved under isomorphism: two matched vertices must have the same degree.
Thus, the degree sequences (monotonic nonincreasing sequence of the vertex degrees) of two graphs
must be identical if they are isomorphic.
Figure 11: The graphs G and H have identical degree sequences, and each have 6 vertices and 6 edges, but
are clearly not isomorphic.
Def: A subgraph G0 of a graph G is a graph formed by a subset of vertices and edges of G.
If two graphs are isomorphic, then subgraphs formed by corresponding vertices and edges must be
isomorphic.
Complete subgraphs are building blocks of all larger graphs. Their identification is often useful in determining whether two graphs are isomorphic.
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Chapter 1: Elements of Graph Theory
Figure 12: Both graphs consist of a K4 and a K2 joined at a common vertex. Once this is noted, isomorphism
of these graphs is easily demonstrated.
Example: Give an isomorphism between the graphs in Figure 12.
Figure 13
Example: Are the graphs in Figure 13 isomorphic?
Def: Given a graph G = (V, E), its complement is a graph G = (V, E) with the same set of vertices but
now with edges between exactly those pairs of vertices not linked in G.
The union of the edges in G and G forms a complete graph.
Two graphs G1 and G2 are isomorphic if and only if G1 and G2 are isomorphic.
(In general, if a graph has more pairs of vertices joined by edges than not, then its complement will have
fewer edges and probability be simpler to analyze.)
Note: when showing that two graphs are not isomorphic, you must give a clear reason. When showing that
two graphs are isomorphic, all is needed is a valid isomorphism.
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Chapter 1: Elements of Graph Theory
Section 1.3: Edge Counting
Theorem 1: In any graph, the sum of the degrees of all vertices is equal to twice the number of edges
P
( x∈V deg(x) = 2 · e).
Proof: Summing the degrees of all vertices counts all instances of some edge being incident at some vertex.
But each edge is incident with two vertices, so the total number of such edge-vertex incidences is simply
twice the number of edges. Corollary: In any graph, the number of vertices of odd degree is even.
Example: Suppose we want to construct a graph with 20 edges and have every vertex of degree 4. How
many vertices must the graph have?
Def: If a graph G is not connected, its vertices can be partitioned into connected pieces, called components.
A component H is a connected subgraph of G such that there is no path between any vertex in H and
any vertex of G not in H.
Since each component of a graph is a graph in its own right, the theorem and corollary above apply to all
components of a graph as well.
Figure 14: Mountain range graph
Example: Consider the mountain range graph in Figure 14. Suppose two people start at locations A and
Z (at the same elevation on opposite sides of the mountain range) and wish to meet at the summit M
by moving in such a way that they are always at the same altitude at every moment. Make a range
graph whose vertices are pairs of points (PL , PR ) at the same altitude with PL on the left side of the
summit and PR on the right side, such that at least one of the two points is a local peak or valley. The
edges of the range graph should join (PL , PR ) and (PL0 , PR0 ) if the two people can move constantly in
the same direction from PL to PL0 and from PR to PR0 , respectively. Is there a path in the range graph
from the starting vertex (A, Z) to the summit vertex (M, M )?
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Chapter 1: Elements of Graph Theory
Example: Find the number of edges in Kn .
Def: The length of a circuit or path is the number of edges in it.
Theorem 2: A graph G is bipartite if and only if every circuit in G has even length.
Proof: Note that is it sufficient to prove this theorem for connected bipartite graphs. (Why?)
(⇒) First we show that if G is bipartite, then any circuit has even length. If G is bipartite, then
it can be drawn with all edges connecting a “left” vertex with a “right” vertex. Thus any circuit
x1 − x2 − x3 − · · · − xn − x1 has alternating left−right−left− · · · −right−left vertices (assuming the first
vertex x1 is on the left). Thus odd-subscripted vertices are on the left and even-subscripted vertices
are on the right. Since xn is adjacent to x1 , xn must be on the right, so its subscript must be even.
Thus there are an even number of (distinct) vertices in the circuit. Any circuit has the same number
of edges as (distinct) vertices, and thus this circuit must have even length.
(⇐) Next we must show that if every circuit in G has even length, then G is bipartite. This direction
of the proof is left as an exercise to the diligent student (see page 28 of the text).
Figure 15
Example: Is the graph in Figure 15 bipartite?
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Chapter 1: Elements of Graph Theory
Section 1.4: Planar Graphs
Def: A graph is planar if it can be drawn on a plane without edges crossing.
Def: The term plane graph refers to a planar depiction of a planar graph.
If a graph G has been drawn with edges crossing, this does not mean the graph is nonplanar. There may
be another way to draw the graph without edges crossing.
Figure 16: Two planar graphs. Note that only the graph on the right is a plane graph. (The graph on the
left can be redrawn in the form of the graph on the right.)
There are two main methods to determining whether a graph is planar. The first approach, called the
circle-chord method, assumes the graph to be planar and attempts to construct a planar representation of the graph—we want to be able to conclude that the graph is not planar if our construction
fails.
The second approach develops some theory to find useful properties of planar graphs.
Circle-Chord Method:
1. Find a circuit that contains all the vertices of the graph. (Note: this step is not always possible,
and it is often difficult.)
2. Draw this circuit as a large circle.
3. Choose one chord (an edge not included in the original circuit), and draw it, say, outside the
circle. Of chosen correctly, it will force certain other chords to be drawn inside the circle.
4. If all chords can be drawn without crossing, the graph is planar and a planar representation has
been found.
Example: Use the circle-chord method to determine if the graph in Figure 17 is planar.
Figure 17
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Chapter 1: Elements of Graph Theory
Figure 18: K3,3 , the complete bipartite graph consisting of two sets of three vertices.
Example: Show that K3,3 is nonplanar using the circle-chord method.
Figure 19: K5 , the complete graph on five vertices.
At-Home Exercise: Show that K5 is nonplanar using the circle-chord method.
Is is possible to prove that any nonplanar graph always contains a K3,3 or a K5 as a subgraph or a slight
modification of these two graphs.
Def: A graph is a K3,3 configuration if it can be obtained from a K3,3 by adding vertices in the middle
of some edges (see Figure 20).
A K5 configuration is defined similarly.
Figure 20: An example of a K3,3 configuration.
Theorem (Kuratowski’s Theorem): A graph is planar if and only if it does not contain a subgraph that
is a K5 or K3,3 configuration.
It is often very difficult to find such a configuration in a nonplanar graph.
A helpful tip: most small nonplanar graphs contain a K3,3 configuration.
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Chapter 1: Elements of Graph Theory
Figure 21
Example: Use the circle-chord method to determine whether the graph in Figure 21 is planar. If it is
nonplanar, find a subgraph that is a K3,3 configuration.
Def: In a plane graph, the edges and vertices of the graph divide the plane into regions, or faces. (Formally,
a region is a maximal section of the plane in which any two points can be joined by a curve that does
not intersect any part of G.)
The unbounded region outside of the graph counts as a region.
Figure 22: The graph on the left has 7 regions (shown on the right).
Let v = # of vertices,
e = # of edges,
r = # of regions.
Theorem (Euler’s Formula): If G is a connected planar graph, then any plane graph depiction of G has
r = e − v + 2 regions.
Proof: (By induction on the number of edges e.) Let G be a connected plane graph with e edges.
If e = 1, then v = 2 and r = 1. Thus, r = 1 = 1 − 2 + 2 = e − v + 2, so the claim holds for the base
case of e = 1.
Now suppose the claim holds for all e < k; we will show that this implies that the claim holds for
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Chapter 1: Elements of Graph Theory
e = k. Since the claim holds for all e < k, it holds for a graph G0 with e = k − 1. Consider adding the
k th edge, (x, y) to G0 to create G.
(i) Suppose the k th edge joins two vertices x and y that are both in G0 . Then both x and y lie on the
boundary of some common region R of G0 . (If x and y did not lie on the boundary of a common
region, edge (x, y) could not be drawn in a planar fashion, as required.) Then, by adding edge
(x, y), K is split into two regions, increasing both e and r by 1. Thus each side of Euler’s formula
grows by 1. Hence, if the formula is true for G0 , it must be true for G.
(ii) Suppose the k th edge joins a vertex in G0 (say, y) to a new vertex (which was not previously in
G) (say, x). Then, by adding edge (x, y), no new regions are formed (since no existing regions
are split). Thus e and v both increase by 1, and r remains the same. Thus each side of Euler’s
formula grows by 1. Hence, if the formula is true for G0 , it must be true for G.
So each increase in r is balanced in Euler’s formula by an increase in e or v. By induction, the formula
is true for any connected planar graph G with e edges. Example: How many regions would there be in a plane graph with 12 vertices each of degree 2?
Corollary: If G is a connected planar graph with e > 1, then e ≤ 3v − 6.
Note that this corollary does not mean that if e ≤ 3v − 6, then the graph is planar. However, the
contrapositive of the corollary is useful for proving a graph is nonplanar: if e > 3v − 6, then G is
nonplanar.
Proof: Let us first define the degree of a region deg(r) to be the number of edges incident to a region (i.e.,
the number of edges on its boundary). If an edge occurs twice along a boundary, the edge is counted
twice in the region’s degree.
Observe that each region in a plane graph must have deg(r) ≥ 3, for a region of degree 2 would be
bounded by two edges joining the same pair of vertices and a region of 1 would be bounded by a loop
edge, but parallel edges and loops are not allowed in graphs.
Since the sum of all regions’ degrees in a graph includes each edge twice, 2e =
P
all regions r
deg(r).
Then, remembering each region must have degree at least 3,
2e =
X
deg(r) ≥ 3r.
all regions r
Thus, 23 e ≥ r. Combining this inequality with Euler’s formula, we have 23 e ≥ e − v + 2. Solving for e,
we obtain
e ≤ 3v − 6.
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Chapter 1: Elements of Graph Theory
Example: Use the corollary to prove that K5 is nonplanar.
Extra Exercises:
1. Is K4 planar? Prove your answer.
2. Prove or provide a counterexample: Every subgraph of a bipartite graph is also bipartite.
3. Prove or provide a counterexample: Every subgraph of a complete graph is also complete.
4. Are the two graphs below isomorphic? Explain carefully.
5. Is the graph below planar? Find a K3,3 or K5 configuration or draw a planar representation.
6. Is the graph below planar? Find a K3,3 or K5 configuration or draw a planar representation.
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