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CS 207m 2018: Discrete Structures (Minor)
Graph theoryBasic terminology, Eulerian walks
Lecture 14Mar 07 2018
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Topic 3: Graph theory
Topics covered in the last three lectures
I Pigeon-Hole Principle and its extensions.
I A glimpse of Ramsey theory
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Topic 3: Graph theory
Topics covered in the last three lectures
I Pigeon-Hole Principle and its extensions.
I A glimpse of Ramsey theory
I Ramsey numbers R(k, `): Existence of regularsub-structures in large enough general structures.
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Topic 3: Graph theory
Topics covered in the last three lectures
I Pigeon-Hole Principle and its extensions.
I A glimpse of Ramsey theory
I Ramsey numbers R(k, `): Existence of regularsub-structures in large enough general structures.
I Structures = Graphs,
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Topic 3: Graph theory
Topics covered in the last three lectures
I Pigeon-Hole Principle and its extensions.
I A glimpse of Ramsey theory
I Ramsey numbers R(k, `): Existence of regularsub-structures in large enough general structures.
I Structures = Graphs, regular= monochromatic/complete
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Topic 3: Graph theory
Topics covered in the last three lectures
I Pigeon-Hole Principle and its extensions.
I A glimpse of Ramsey theory
I Ramsey numbers R(k, `): Existence of regularsub-structures in large enough general structures.
I Structures = Graphs, regular= monochromatic/complete
Next topic
Graphs and their properties!
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Topic 3: Graph theory
Textbook Reference
I Introduction to Graph Theory, 2nd Ed., by Douglas West.
I Low cost Indian edition available, published by PHILearning Private Ltd.
Topics covered in the next two lectures:
I What is a Graph?
I Paths, cycles, walks and trails; connected graphs.
I Eulerian graphs and a characterization in terms of degreesof vertices.
I Bipartite graphs and a characterization in terms of oddlength cycles.
Reference: Section 1.1, 1.2 of Chapter 1 from Douglas West.
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Topic 3: Graph theory
Textbook Reference
I Introduction to Graph Theory, 2nd Ed., by Douglas West.
I Low cost Indian edition available, published by PHILearning Private Ltd.
Topics covered in the next two lectures:
I What is a Graph?
I Paths, cycles, walks and trails; connected graphs.
I Eulerian graphs and a characterization in terms of degreesof vertices.
I Bipartite graphs and a characterization in terms of oddlength cycles.
Reference: Section 1.1, 1.2 of Chapter 1 from Douglas West.
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What are graphs
Recall:
Definition
A simple graph G is a pair (V,E) of a set of vertices/nodes Vand edges E between unordered pairs of vertices calledend-points: e = vu means that e is an edge between v and u(u 6= v).
I What about loops?I What about directed edges?I What about multiple edges?
General Definition
A graph G is a triple V,E,R where V is a set of vertices, E is aset of edges and R ⊆ E × V × V is a relation that associateseach edge with two vertices called its end-points.
We will consider only finite graphs (i.e., |V |, |E| are finite) andoften deal with simple graphs.
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What are graphs
Recall:
Definition
A simple graph G is a pair (V,E) of a set of vertices/nodes Vand edges E between unordered pairs of vertices calledend-points: e = vu means that e is an edge between v and u(u 6= v).
I What about loops?I What about directed edges?I What about multiple edges?
General Definition
A graph G is a triple V,E,R where V is a set of vertices, E is aset of edges and R ⊆ E × V × V is a relation that associateseach edge with two vertices called its end-points.
We will consider only finite graphs (i.e., |V |, |E| are finite) andoften deal with simple graphs.
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What are graphs
Recall:
Definition
A simple graph G is a pair (V,E) of a set of vertices/nodes Vand edges E between unordered pairs of vertices calledend-points: e = vu means that e is an edge between v and u(u 6= v).
I What about loops?I What about directed edges?I What about multiple edges?
General Definition
A graph G is a triple V,E,R where V is a set of vertices, E is aset of edges and R ⊆ E × V × V is a relation that associateseach edge with two vertices called its end-points.
We will consider only finite graphs (i.e., |V |, |E| are finite) andoften deal with simple graphs.
4
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Konigsberg Bridge problem
I In 18th century Prussia, the city on river Pregel...
I Question was to find a walk from home, crossing everybridge exactly once and returning home.
I Leonard Euler showed in 1735 that this is impossible.
I Thus, he “gave birth” to the area of graph theory.
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Konigsberg Bridge problem
I In 18th century Prussia, the city on river Pregel...
I Question was to find a walk from home, crossing everybridge exactly once and returning home.
I Leonard Euler showed in 1735 that this is impossible.
I Thus, he “gave birth” to the area of graph theory.
5
![Page 14: CS 207m 2018: Discrete Structures (Minor)akshayss/courses/cs207m-2018/... · 2018. 3. 8. · Topic 3: Graph theory Textbook Reference I Introduction to Graph Theory, 2nd Ed., by Douglas](https://reader035.vdocument.in/reader035/viewer/2022071421/611ab065ceb1b711d2531f6a/html5/thumbnails/14.jpg)
Konigsberg Bridge problem
I In 18th century Prussia, the city on river Pregel...
I Question was to find a walk from home, crossing everybridge exactly once and returning home.
I Leonard Euler showed in 1735 that this is impossible.
I Thus, he “gave birth” to the area of graph theory.
5
![Page 15: CS 207m 2018: Discrete Structures (Minor)akshayss/courses/cs207m-2018/... · 2018. 3. 8. · Topic 3: Graph theory Textbook Reference I Introduction to Graph Theory, 2nd Ed., by Douglas](https://reader035.vdocument.in/reader035/viewer/2022071421/611ab065ceb1b711d2531f6a/html5/thumbnails/15.jpg)
Konigsberg Bridge problem
I Qn: Find a walk from home, which crosses every bridgeexactly once and returns home.
I Leonard Euler showed in 1735 that this is impossible. Theargument is as follows:
I Each time you enter or leave a vertex, you use an edge.I So each vertex must be connected to an even no. of vertices.I Which is not the case here, hence it is impossible.
I Clearly, this is a sufficient condition, but is it necessary?I If every vertex is connected to an even no. of vertices in a
graph, is there such a walk? This is called Eulerian walk.
6
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Konigsberg Bridge problem
I Qn: Find a walk from home, which crosses every bridgeexactly once and returns home.
I Leonard Euler showed in 1735 that this is impossible. Theargument is as follows:
I Each time you enter or leave a vertex, you use an edge.
I So each vertex must be connected to an even no. of vertices.I Which is not the case here, hence it is impossible.
I Clearly, this is a sufficient condition, but is it necessary?I If every vertex is connected to an even no. of vertices in a
graph, is there such a walk? This is called Eulerian walk.
6
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Konigsberg Bridge problem
I Qn: Find a walk from home, which crosses every bridgeexactly once and returns home.
I Leonard Euler showed in 1735 that this is impossible. Theargument is as follows:
I Each time you enter or leave a vertex, you use an edge.I So each vertex must be connected to an even no. of vertices.
I Which is not the case here, hence it is impossible.I Clearly, this is a sufficient condition, but is it necessary?I If every vertex is connected to an even no. of vertices in a
graph, is there such a walk? This is called Eulerian walk.
6
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Konigsberg Bridge problem
I Qn: Find a walk from home, which crosses every bridgeexactly once and returns home.
I Leonard Euler showed in 1735 that this is impossible. Theargument is as follows:
I Each time you enter or leave a vertex, you use an edge.I So each vertex must be connected to an even no. of vertices.I Which is not the case here, hence it is impossible.
I Clearly, this is a sufficient condition, but is it necessary?I If every vertex is connected to an even no. of vertices in a
graph, is there such a walk? This is called Eulerian walk.
6
![Page 19: CS 207m 2018: Discrete Structures (Minor)akshayss/courses/cs207m-2018/... · 2018. 3. 8. · Topic 3: Graph theory Textbook Reference I Introduction to Graph Theory, 2nd Ed., by Douglas](https://reader035.vdocument.in/reader035/viewer/2022071421/611ab065ceb1b711d2531f6a/html5/thumbnails/19.jpg)
Konigsberg Bridge problem
I Qn: Find a walk from home, which crosses every bridgeexactly once and returns home.
I Leonard Euler showed in 1735 that this is impossible. Theargument is as follows:
I Each time you enter or leave a vertex, you use an edge.I So each vertex must be connected to an even no. of vertices.I Which is not the case here, hence it is impossible.
I Clearly, this is a sufficient condition, but is it necessary?
I If every vertex is connected to an even no. of vertices in agraph, is there such a walk? This is called Eulerian walk.
6
![Page 20: CS 207m 2018: Discrete Structures (Minor)akshayss/courses/cs207m-2018/... · 2018. 3. 8. · Topic 3: Graph theory Textbook Reference I Introduction to Graph Theory, 2nd Ed., by Douglas](https://reader035.vdocument.in/reader035/viewer/2022071421/611ab065ceb1b711d2531f6a/html5/thumbnails/20.jpg)
Konigsberg Bridge problem
I Qn: Find a walk from home, which crosses every bridgeexactly once and returns home.
I Leonard Euler showed in 1735 that this is impossible. Theargument is as follows:
I Each time you enter or leave a vertex, you use an edge.I So each vertex must be connected to an even no. of vertices.I Which is not the case here, hence it is impossible.
I Clearly, this is a sufficient condition, but is it necessary?I If every vertex is connected to an even no. of vertices in a
graph, is there such a walk?
This is called Eulerian walk.
6
![Page 21: CS 207m 2018: Discrete Structures (Minor)akshayss/courses/cs207m-2018/... · 2018. 3. 8. · Topic 3: Graph theory Textbook Reference I Introduction to Graph Theory, 2nd Ed., by Douglas](https://reader035.vdocument.in/reader035/viewer/2022071421/611ab065ceb1b711d2531f6a/html5/thumbnails/21.jpg)
Konigsberg Bridge problem
I Qn: Find a walk from home, which crosses every bridgeexactly once and returns home.
I Leonard Euler showed in 1735 that this is impossible. Theargument is as follows:
I Each time you enter or leave a vertex, you use an edge.I So each vertex must be connected to an even no. of vertices.I Which is not the case here, hence it is impossible.
I Clearly, this is a sufficient condition, but is it necessary?I If every vertex is connected to an even no. of vertices in a
graph, is there such a walk? This is called Eulerian walk.6
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Basic terminology
The degree d(v) of a vertex v (in an undirected loopless graph)is the number of edges incident to it, i.e., |{vw ∈ E | w ∈ V }|.A vertex of degree 0 is called an isolated vertex.
I A walk is a sequence of vertices v1, . . . vk such that∀i ∈ {1, . . . k− 1}, (vi, vi+1) ∈ E. The vertices v1 and vk arecalled the end-points and others are called internal vertices.
I A path is a walk in which no vertex is repeated. Its lengthis the number of edges in it.
I A walk is called closed if it starts and ends with the samevertex, i.e., its endpoints are the same.
I A closed walk is called a cycle if its internal vertices are alldistinct from each other and from the end-point.
A graph is called connected if there is a path (or walk) betweenany two of its vertices.
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Basic terminology
The degree d(v) of a vertex v (in an undirected loopless graph)is the number of edges incident to it, i.e., |{vw ∈ E | w ∈ V }|.A vertex of degree 0 is called an isolated vertex.
I A walk is a sequence of vertices v1, . . . vk such that∀i ∈ {1, . . . k− 1}, (vi, vi+1) ∈ E. The vertices v1 and vk arecalled the end-points and others are called internal vertices.
I A path is a walk in which no vertex is repeated. Its lengthis the number of edges in it.
I A walk is called closed if it starts and ends with the samevertex, i.e., its endpoints are the same.
I A closed walk is called a cycle if its internal vertices are alldistinct from each other and from the end-point.
A graph is called connected if there is a path (or walk) betweenany two of its vertices.
7
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Basic terminology
The degree d(v) of a vertex v (in an undirected loopless graph)is the number of edges incident to it, i.e., |{vw ∈ E | w ∈ V }|.A vertex of degree 0 is called an isolated vertex.
I A walk is a sequence of vertices v1, . . . vk such that∀i ∈ {1, . . . k− 1}, (vi, vi+1) ∈ E. The vertices v1 and vk arecalled the end-points and others are called internal vertices.
I A path is a walk in which no vertex is repeated. Its lengthis the number of edges in it.
I A walk is called closed if it starts and ends with the samevertex, i.e., its endpoints are the same.
I A closed walk is called a cycle if its internal vertices are alldistinct from each other and from the end-point.
A graph is called connected if there is a path (or walk) betweenany two of its vertices.
7
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Basic terminology
The degree d(v) of a vertex v (in an undirected loopless graph)is the number of edges incident to it, i.e., |{vw ∈ E | w ∈ V }|.A vertex of degree 0 is called an isolated vertex.
I A walk is a sequence of vertices v1, . . . vk such that∀i ∈ {1, . . . k− 1}, (vi, vi+1) ∈ E. The vertices v1 and vk arecalled the end-points and others are called internal vertices.
I A path is a walk in which no vertex is repeated. Its lengthis the number of edges in it.
I A walk is called closed if it starts and ends with the samevertex, i.e., its endpoints are the same.
I A closed walk is called a cycle if its internal vertices are alldistinct from each other and from the end-point.
A graph is called connected if there is a path (or walk) betweenany two of its vertices.
7
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Basic terminology
The degree d(v) of a vertex v (in an undirected loopless graph)is the number of edges incident to it, i.e., |{vw ∈ E | w ∈ V }|.A vertex of degree 0 is called an isolated vertex.
I A walk is a sequence of vertices v1, . . . vk such that∀i ∈ {1, . . . k− 1}, (vi, vi+1) ∈ E. The vertices v1 and vk arecalled the end-points and others are called internal vertices.
I A path is a walk in which no vertex is repeated. Its lengthis the number of edges in it.
I A walk is called closed if it starts and ends with the samevertex, i.e., its endpoints are the same.
I A closed walk is called a cycle if its internal vertices are alldistinct from each other and from the end-point.
A graph is called connected if there is a path (or walk) betweenany two of its vertices.
7
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Basic terminology
The degree d(v) of a vertex v (in an undirected loopless graph)is the number of edges incident to it, i.e., |{vw ∈ E | w ∈ V }|.A vertex of degree 0 is called an isolated vertex.
I A walk is a sequence of vertices v1, . . . vk such that∀i ∈ {1, . . . k− 1}, (vi, vi+1) ∈ E. The vertices v1 and vk arecalled the end-points and others are called internal vertices.
I A path is a walk in which no vertex is repeated. Its lengthis the number of edges in it.
I A walk is called closed if it starts and ends with the samevertex, i.e., its endpoints are the same.
I A closed walk is called a cycle if its internal vertices are alldistinct from each other and from the end-point.
A graph is called connected if there is a path (or walk) betweenany two of its vertices.
7
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Basic terminology: some exercises
v1 v2 v3
v4 v5 v6
v7 v8
1. Give examples each of the following in the above graph:1.1 All vertices of degree 31.2 A walk of length 51.3 A path of length 51.4 A closed walk of length 61.5 A cycle of length 6
2. True or False: (a) If a graph is not connected, it must havean isolated vertex.(b) A graph is connected iff some vertex is connected toevery other vertex.
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Basic terminology: some exercises
v1 v2 v3
v4 v5 v6
v7 v8
1. Give examples each of the following in the above graph:1.1 All vertices of degree 31.2 A walk of length 51.3 A path of length 51.4 A closed walk of length 61.5 A cycle of length 6
2. True or False: (a) If a graph is not connected, it must havean isolated vertex.(b) A graph is connected iff some vertex is connected toevery other vertex.
8
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Eulerian graphs
Definition
A graph is called Eulerian if it has a closed walk that containsall edges, and each edge occurs exactly once. Such a walk iscalled an Eulerian walk.
I If G is Eulerian, each vertex must have an even degree.I This is a necessary condition, but is it sufficient?
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof: ( =⇒ )I Suppose G is Eulerian: every vertex has even degree.
I each passage through a vertex uses two edges (in and out).I at the first vertex first edge is paired with last.
I Any two edges are in the same walk implies graph isconnected (unless it has isolated vertices).
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Eulerian graphs
Definition
A graph is called Eulerian if it has a closed walk that containsall edges, and each edge occurs exactly once. Such a walk iscalled an Eulerian walk.
I If G is Eulerian, each vertex must have an even degree.I This is a necessary condition, but is it sufficient?
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof: ( =⇒ )I Suppose G is Eulerian: every vertex has even degree.
I each passage through a vertex uses two edges (in and out).I at the first vertex first edge is paired with last.
I Any two edges are in the same walk implies graph isconnected (unless it has isolated vertices).
9
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Eulerian graphs
Definition
A graph is called Eulerian if it has a closed walk that containsall edges, and each edge occurs exactly once. Such a walk iscalled an Eulerian walk.
I If G is Eulerian, each vertex must have an even degree.I This is a necessary condition, but is it sufficient?
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof: ( =⇒ )I Suppose G is Eulerian: every vertex has even degree.
I each passage through a vertex uses two edges (in and out).I at the first vertex first edge is paired with last.
I Any two edges are in the same walk implies graph isconnected (unless it has isolated vertices).
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Eulerian graphs
Definition
A graph is called Eulerian if it has a closed walk that containsall edges, and each edge occurs exactly once. Such a walk iscalled an Eulerian walk.
I If G is Eulerian, each vertex must have an even degree.I This is a necessary condition, but is it sufficient?
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof:
( =⇒ )I Suppose G is Eulerian: every vertex has even degree.
I each passage through a vertex uses two edges (in and out).I at the first vertex first edge is paired with last.
I Any two edges are in the same walk implies graph isconnected (unless it has isolated vertices).
9
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Eulerian graphs
Definition
A graph is called Eulerian if it has a closed walk that containsall edges, and each edge occurs exactly once. Such a walk iscalled an Eulerian walk.
I If G is Eulerian, each vertex must have an even degree.I This is a necessary condition, but is it sufficient?
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof: ( =⇒ )I Suppose G is Eulerian: every vertex has even degree.
I each passage through a vertex uses two edges (in and out).I at the first vertex first edge is paired with last.
I Any two edges are in the same walk implies graph isconnected (unless it has isolated vertices).
9
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Eulerian graphs
Definition
A graph is called Eulerian if it has a closed walk that containsall edges, and each edge occurs exactly once. Such a walk iscalled an Eulerian walk.
I If G is Eulerian, each vertex must have an even degree.I This is a necessary condition, but is it sufficient?
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof: ( =⇒ )I Suppose G is Eulerian: every vertex has even degree.
I each passage through a vertex uses two edges (in and out).I at the first vertex first edge is paired with last.
I Any two edges are in the same walk implies graph isconnected (unless it has isolated vertices).
9
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Maximal paths
I A path P is said to be maximal in G if it is not containedin any longer path.
I If a graph is finite, no path can extend forever, so maximal(non-extendible) paths exist.
I A maximum path is a path of maximum length among allpaths in the graph.Is every maximal path maximum?
v1
v2
v3 v4
I List all the maximal and maximum paths in the abovegraph.
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Maximal paths
I A path P is said to be maximal in G if it is not containedin any longer path.
I If a graph is finite, no path can extend forever, so maximal(non-extendible) paths exist.
I A maximum path is a path of maximum length among allpaths in the graph.
Is every maximal path maximum?
v1
v2
v3 v4
I List all the maximal and maximum paths in the abovegraph.
10
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Maximal paths
I A path P is said to be maximal in G if it is not containedin any longer path.
I If a graph is finite, no path can extend forever, so maximal(non-extendible) paths exist.
I A maximum path is a path of maximum length among allpaths in the graph.Is every maximal path maximum?
v1
v2
v3 v4
I List all the maximal and maximum paths in the abovegraph.
10
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Maximal paths
I A path P is said to be maximal in G if it is not containedin any longer path.
I If a graph is finite, no path can extend forever, so maximal(non-extendible) paths exist.
I A maximum path is a path of maximum length among allpaths in the graph.Is every maximal path maximum?
v1
v2
v3 v4
I List all the maximal and maximum paths in the abovegraph.
10
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Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
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Eulerian graphs
A path P is said to be maximal in G if it is not contained inany longer path. In a finite graph maximal paths exist.
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
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Eulerian graphs
A path P is said to be maximal in G if it is not contained inany longer path. In a finite graph maximal paths exist.
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Proof:
u v
I Let P be a maximal path in G. It starts from some u.
11
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Eulerian graphs
A path P is said to be maximal in G if it is not contained inany longer path. In a finite graph maximal paths exist.
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Proof:
u v
I Let P be a maximal path in G. It starts from some u.
I Since P cannot be extended, every nbr of u is already in P .
11
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Eulerian graphs
A path P is said to be maximal in G if it is not contained inany longer path. In a finite graph maximal paths exist.
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Proof:
u v
I Let P be a maximal path in G. It starts from some u.
I Since P cannot be extended, every nbr of u is already in P .
I As d(u) ≥ 2,∃v in P such that uv 6∈ P .
I Thus we have the cycle u . . . vu.
11
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Eulerian graphs
A path P is said to be maximal in G if it is not contained inany longer path. In a finite graph maximal paths exist.
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Proof:u v
I Let P be a maximal path in G. It starts from some u.I Since P cannot be extended, every nbr of u is already in P .I As d(u) ≥ 2,∃v in P such that uv 6∈ P .I Thus we have the cycle u . . . vu.
Is this true if G were infinite?
No! Consider V = Z, E = {ij : |i− j| = 1}.−2 −1 0 1 2 3
· · ·· · ·
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Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Proof:
u v
I Let P be a maximal path in G. It starts from some u.
I Since P cannot be extended, every nbr of u is already in P .
I As d(u) ≥ 2,∃v in P such that uv 6∈ P .
I Thus we have the cycle u . . . vu.
Is this true if G were infinite?No! Consider V = Z, E = {ij : |i− j| = 1}.
−2 −1 0 1 2 3
· · ·· · ·
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Proof of theorem on Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
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Proof of theorem on Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof (⇐=): By induction on number of edges m.
I Base case: m = 1.
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Proof of theorem on Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof (⇐=): By induction on number of edges m.
I Induction step: m > 1. By Lemma G has a cycle C.
12
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Proof of theorem on Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof (⇐=): By induction on number of edges m.
I Induction step: m > 1. By Lemma G has a cycle C.I Delete all edges in cycle C and (new) isolated vertices.
12
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Proof of theorem on Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof (⇐=): By induction on number of edges m.
I Induction step: m > 1. By Lemma G has a cycle C.I Delete all edges in cycle C and (new) isolated vertices.I Get G1, . . . , Gk. Each Gi is
I connectedI has < m edges.I all its vertices have even degree (why? )
12
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Proof of theorem on Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof (⇐=): By induction on number of edges m.
I Induction step: m > 1. By Lemma G has a cycle C.I Delete all edges in cycle C and (new) isolated vertices.I Get G1, . . . , Gk. Each Gi is
I connectedI has < m edges.I all its vertices have even degree (why? – degree of any
vertex was even and removing C, reduces each vertexdegree by 0 or 2.)
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Proof of theorem on Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof (⇐=): By induction on number of edges m.
I Induction step: m > 1. By Lemma G has a cycle C.I Delete all edges in cycle C and (new) isolated vertices.I By indn hyp, each Gi is Eulerian, has a Eulerian walk.
12
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Proof of theorem on Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof (⇐=): By induction on number of edges m.
I Induction step: m > 1. By Lemma G has a cycle C.I Delete all edges in cycle C and (new) isolated vertices.I By indn hyp, each Gi is Eulerian, has a Eulerian walk.I Combine the cycle C and the Eulerian walk of G1, . . . , Gk
to produce an Eulerian walk of G (how?).
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Proof of theorem on Eulerian graphs
Lemma
If every vertex of a graph G has degree at least 2, then Gcontains a cycle.
Theorem
A graph G with no isolated vertices is Eulerian iff it isconnected and all its vertices have even degree.
Proof (⇐=): By induction on number of edges m.I Induction step: m > 1. By Lemma G has a cycle C.
I Delete all edges in cycle C and (new) isolated vertices.I By indn hyp, each Gi is Eulerian, has a Eulerian walk.I Combine the cycle C and the Eulerian walk of G1, . . . , Gk
to produce an Eulerian walk of G (how?).I Traverse along cycle C in G and when some Gi is entered
for first time, detour along an Eulerian walk of Gi.I This walk ends at vertex where we started detour.I When we complete traversal of C in this way, we have
completed an Eulerian walk on G.12
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Principle of extremality
Principle of extremality
I In proving the lemma we used a “new” important prooftechnique, called extremality.
I By considering some “extreme” structure, we got someadditional information which we used in the proof.
I E.g., Above, since a maximal path could not be extended,we got that every neighbour of an endpoint of P is in P .
I (H.W) Can you show the theorem directly fromextremality without using induction?
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Principle of extremality
Principle of extremality
I In proving the lemma we used a “new” important prooftechnique, called extremality.
I By considering some “extreme” structure, we got someadditional information which we used in the proof.
I E.g., Above, since a maximal path could not be extended,we got that every neighbour of an endpoint of P is in P .
I (H.W) Can you show the theorem directly fromextremality without using induction?
13
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Principle of extremality
Principle of extremality
I In proving the lemma we used a “new” important prooftechnique, called extremality.
I By considering some “extreme” structure, we got someadditional information which we used in the proof.
I E.g., Above, since a maximal path could not be extended,we got that every neighbour of an endpoint of P is in P .
I (H.W) Can you show the theorem directly fromextremality without using induction?
13
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Principle of extremality
Principle of extremality
I In proving the lemma we used a “new” important prooftechnique, called extremality.
I By considering some “extreme” structure, we got someadditional information which we used in the proof.
I E.g., Above, since a maximal path could not be extended,we got that every neighbour of an endpoint of P is in P .
I (H.W) Can you show the theorem directly fromextremality without using induction?
13
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Applications of Eulerian graphs
Corollary
Every graph with all vertices having even degree decomposesinto cycles
Proof: (H.W) or read from Douglas West’s book!
An interesting application of Eulerian graphs
If we want to draw a given connected graph G on paper, howmany times must we stop and move the pen? No segmentshould be drawn twice.
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Applications of Eulerian graphs
Corollary
Every graph with all vertices having even degree decomposesinto cycles
Proof: (H.W) or read from Douglas West’s book!
An interesting application of Eulerian graphs
If we want to draw a given connected graph G on paper, howmany times must we stop and move the pen? No segmentshould be drawn twice.
14
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Applications of Eulerian graphs
Corollary
Every graph with all vertices having even degree decomposesinto cycles
Proof: (H.W) or read from Douglas West’s book!
An interesting application of Eulerian graphs
If we want to draw a given connected graph G on paper, howmany times must we stop and move the pen? No segmentshould be drawn twice.
14