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- #MA284 - Discrete Mathematics
- Previous Topic: Definitions & Planar Graphs
- Next Topic: Colouring Graphs; Eulerian & Hamiltonian Graphs
- Relevant Slides:
-
Non-Planar Graphs
- Most graphs do not have a planar representation, however, it takes some work to prove that a graph is non-planar.
- To do this, we can use Euler's formula for planar graphs to prove that they are not planar.
-
Theorem:
K_{5}
is not planar (Theorem 4.3.1 in Textbook)- The proof is by contradiction.
- So assume that
K_5
is planar. Then, the graph must satisfy Euler's formula for planar graphs.K_5
has 5 vertices & 10 edges, so we get5-10 + f =2
, which says that if the graph is drawn without any edges crossing, there would bef = 7
faces.
- Now consider how many edges surround each face. Each face must be surrounded by at least 3 edges.
- Let
B
be the total number of boundaries around all the faces in the graph.- Thus, we have that
3f \leq B
, but alsoB = 2e
, as each edge is used as a boundary exactly twice.- Putting this together, we get
3f \leq B
, but this is impossible, since we have already determined thatf = 7
ande = 10
, and21 \nleq 20
.- This is a contradiction, so
K_5
is not planar.
- This is a contradiction, so
- Putting this together, we get
- Thus, we have that
- Let
- Q.E.D.
-
Theorem:
K_{3,3}
is not planar (Theorem 4.2.2 in Textbook)- Please read the proof in the textbook (or edit this later).
- The proof for
K_{3,3}
is somewhat similar toK_5
, but it also uses the fact that a bipartite graph has no 3-edge cycles.
- To understand the importance of
K_5
&K_{3,3}
, we first need that the concept of homeomorphic graphs.- Recall that a graph
G_1
is a subgraph ofG
if it can be obtained by deleting some vertices and / or edges ofG
.
- Recall that a graph
-
Homeomorphic Graphs
- What is a subdivision of an edge? #card
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- A subdivision of an edge is obtained by "adding" a new vertex of degree 2 to the middle of the edge.
- What is a subdivision of a graph? #card
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- A subdivision of a graph is obtained by subdividing one or more of its edges.
- What is the smoothing of a pair of edges? #card
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- The smoothing of the pair of edges
\{a,b\}
&\{b,c\}
, whereb
is a vertex of degree 2, means to remove these two edges, and add\{a,c\}
.
- The smoothing of the pair of edges
- What does it mean if two graphs are homeomorphic? #card
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- Two graphs
G_1
,G_2
are homeomorphic if there is some subdivision ofG_1
that is isomorphic to some subdivision ofG_2
.
- Two graphs
-
Kuratowski's Theorem #card
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- A graph is planar if and only if it does not contain a subgraph that is homeomorphic to
K_5
orK_{3,3}
.
- A graph is planar if and only if it does not contain a subgraph that is homeomorphic to
- What this really means is that every non-planar graph has some smoothing that contains a copy of
K_5
orK_{3,3}
somewhere inside it.
- What is Kuratowski's Theorem? #card
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- What is a subdivision of an edge? #card
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- Most graphs do not have a planar representation, however, it takes some work to prove that a graph is non-planar.
-
Polyhedra
- What is a polyhedron? #card
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- A polyhedron is a geometric solid made up of flat polygonal faces joined at edges & vertices.
- What is a convex polyhedron? #card
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- A convex polyhedron is one where any line segment connecting two points on the interior of the polyhedron must be entirely contained inside the polyhedron.
- A remarkable, and important fact, is that every convex polyhedron can be porjected onto the plane without edges crossing.
- Now that we know that every convex polyhedron can be represented as a planar graph, we can apply Euler's formula.
-
Euler's Formula for Polyhedra #card
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The Handshaking Lemma #card
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2|E|
:- Let
G = (V,E)
be a graph, with verticesV = v_1, v_2, \cdots, v_n
. - Let
\text{deg}(v_i)
be the "degree of $v+i$". Then-
\text{deg}(v_1) + \text{deg}(v_2) + \cdots + \text{deg}(v_n) = 2|E|
-
- Let
- The sum of the vertex degrees is
-
-
Example
- Show that there is no convex polyhedron with 11 vertices, all of degree 3. background-color:: green
- If such a convex polyhedron existed, we could draw its graph.
- Each vertex has degree 3, so
-
2|E| = \sum d(v_i) = 3 \times 11
-
\therefore |E| = \frac{33}{2} \text{, which is impossible}
-
- Each vertex has degree 3, so
- What is a polyhedron? #card
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-
The Platonic Solids
- What is a Regular Polyhedron? #card
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- A polyhedron is called regular if:
- All its faces are identical, regular polygons.
- All its vertices have the same degree.
- A polyhedron is called regular if:
- What are the Platonic Solids? #card
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- The convex regular polyhedra are also called the Platonic Solids.
- There are exactly 5 regular polyhedra. This fact can be proven using Euler's formula.
- For full details, see the proof in the textbook.
- Here is the basic idea:
- Consider a regular polyhedron with
f
triangular faces.- So
2e - 3f
.
- So
- Suppose that every vertex has degree
k
.- So
2e - vk
. Also,v - e +f =2
. So-
e = \frac{3f}{2}, v = \frac{3f}{k}, \Rightarrow \frac{3f}{k} - \frac{3f}{2} + f =2
- and thus
-
f = \frac{4k}{6-k}
f
is defined for anyk < 6
, but undefined fork = 6
, and ifk > 6
thenf < 0
- no solutions.
-
-
- So
- Consider a regular polyhedron with
- What is a Regular Polyhedron? #card
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