Graph Theory Summary Notes

CS1231 Graph Theory Summary Notes

CS1231 Discrete Structure

Chapter 11 Graph Theory Summary Notes

Contents Section 1 Terminology Undirected Graph

Vertex Degree The Handshake Theorem

Directed Graph

Section 2 Connectivity

Complete Graphs

Section 3 Euler Tour

Complete Bipartite Graph

Section 4 Hamilton Cycles

Subgraphs

Section 5 Graph Representation

Last Edit: 18th October 2014

Section 1 Terminology Undirected Graph 1. A pseudograph 𝐺 = (𝑉(𝐺), 𝐸(𝐺), 𝑓𝐺 ) consists of 

a non-empty vertex set 𝑉(𝐺) of vertices



an edge set 𝐸(𝐺) of edges



an incidence function 𝑓𝐺 : 𝐸(𝐺) → {{𝑢, 𝑣} | 𝑢, 𝑣 ∈ 𝑉(𝐺)}. [NOTE: vertex set 𝑉(𝐺) must be non-empty]

2. Edges are undirected in an undirected graph. 3. An edge e is a loop if 𝑓𝐺 (ⅇ) = {𝑢, 𝑢} = {𝑢} for some 𝑢 ∈ 𝑉(𝐺). 4. Two distinct edges ⅇ1 and ⅇ2 are multiple (or parallel) edges if 𝑓𝐺 (ⅇ1 ) = 𝑓𝐺 (ⅇ2 ). 5. A simple graph is a pseudograph with no loops and parallel edges 6. A multigraph is a pseudograph with no loops. 7. An edge e is incident with vertices u and v (vice versa) if 𝑓𝐺 (ⅇ) = {𝑢, 𝑣}. The edge e is said to connect its endpoints u and v. 8. Two vertices are adjacent (or neighbours) if they are incident with a common edge. 9. Two edges are called adjacent if they are incident with a common vertex.

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CS1231 Graph Theory Summary Notes

Directed Graph 10. A directed multigraph 𝐺 = (𝑉(𝐷), 𝐸(𝐷), 𝑓𝐷 ) consists of 

a non-empty vertex set 𝑉(𝐷) of vertices



an edge set 𝐸(𝐷) of edges



an incidence function 𝑓𝐷 : 𝐸(𝐷) → {(𝑢, 𝑣)| 𝑢, 𝑣 ∈ 𝑣(𝐷)} [NOTE: difference with undirected graph here is the use of notation of round brackets instead of curly brackets  ordered pair is used  indicate the importance of sequence]

11. The directed edge e start at initial vertex u and end at terminal (or end) vertex v if 𝑓𝐷 (ⅇ) = (𝑢, 𝑣). 12. A directed edge e is a loop if 𝑓𝐷 (ⅇ) = (𝑢, 𝑢) for some 𝑢 ∈ 𝑉(𝐷). 13. The directed edges e1 and e2 are multiple directed edges if 𝑓𝐷 (ⅇ1 ) = 𝑓𝐷 (ⅇ2 ). 14. A simple directed graph is a directed multigraph with no loops and multiple directed edges.

Complete Graphs 15. A complete graph on n vertices, denoted by Kn, is a simple graph in which every two distinct vertices area adjacent. 𝑛 Number of edges in complete graph Kn = |𝐸(𝑘𝑛 )| = ( ) = 2

𝑛(𝑛−1) 2

Complete Bipartite Graph 16. A complete bipartite graph on (m,n) vertices, denoted b Km,n, is a simple graph with 

𝑉(𝐾𝑚,𝑛 ) = {𝑢1 , 𝑢2 , . . . , 𝑢𝑚 } ∪ {𝑣1 , 𝑣2 … 𝑣𝑛 }



𝐸(𝐾𝑚,𝑛 ) = { {𝑢𝑖′ 𝑣𝑗 } | ⅈ = 1 … 𝑚; 𝑗 = 1 … 𝑛 } Number of edges between u vertices on the top or bottom = |𝐸(𝑘𝑚,𝑛 )| = 𝑚 × 𝑛

Subgraphs 17. A graph H is a subgraph of a graph G if 

𝑉(𝐻) ⊆ 𝑉(𝐺)



𝐸(𝐻) ⊆ 𝐸(𝐺)



∀ⅇ ∈ 𝐸(𝐻) (𝑓𝐻 (ⅇ) = 𝑓𝐺 (ⅇ))

Vertex Degree 18. The degree 𝑑𝐺 (𝑣) of a vertex v in an undirected graph G is the number if edges incident with v, each loop counting as 2 edges. A vertex of degree 0 is isolated. YANG LU

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CS1231 Graph Theory Summary Notes

The Handshake Theorem THEOREM 1 Let G be an undirected graph. Then

∑ 𝑑𝐺 (𝑣) = 2|𝐸(𝐺)| 𝑣∈𝑉(𝐺)

COROLLARY 1 Sum of degrees of all vertices of G is even.

PROOF: Combinational Proof 1) List each edge and label its endpoints with the vertex names. 2) The number of times each vertex name is used is the vertex degree. 3) All vertex names appear 2|𝐸(𝐺)| times, by STEP 1. 4) This is also the sum of all the vertex degrees, by STEP 2. Q.E.D.

COROLLARY 2 In an undirected graph, the number of vertices of odd degree is even.

PROOF: Direct Proof 1) Let V1 and V2 be the sets of vertices of odd and even degree in G, respectively. 2) Therefore, ∑ 𝑑𝐺 (𝑣) + ∑ 𝑑𝐺 (𝑣) = 𝑣∈𝑉1

𝑣∈𝑉2

∑ 𝑑𝐺 (𝑣) 𝑣∈𝑉(𝐺)

3) Therefore RHS is even, by Theorem 1. 4) Therefore LHS is even by STEP 3. 5) ∑𝑉∈𝑉1 𝑑𝐺 (𝑣) is even. 6) Therefore ∑𝑣∈𝑉1 𝑑𝐺 (𝑣) = ∑𝑣∈𝑉 (𝑑𝐺 (𝑣) − 1) + ∑𝑣∈𝑉1 1 and |𝑉1 | = ∑𝑣∈𝑉1 1, |𝑉1 | is even. 1

Q.E.D.

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CS1231 Graph Theory Summary Notes

Section 2 Connectivity 1. A walk of length n in an undirected graph G is a finite alternating sequence of vertices and edges of G such that, 𝑣0 ⅇ1 𝑣1 ⅇ2 … 𝑣𝑛−1 ⅇ𝑛 𝑣𝑛 

ⅇ𝑖 connects its endpoints 𝑣𝑖−1 and 𝑣𝑖 for I = 1, … n



vertices vo and vn are the origin and terminus respectively



v1 … vn-1 are internal vertices



if in a simple graph a walk can be specified uniquely by its vertex sequence

2. A trail is a walk with distinct edges. [no repeated edges] 3. A path is a trail with distinct vertice. [no repeated vertices except for origin or terminus] 4. An undirected graph G is connected if there is a walk between every pair of distinct vertices of G. 5. An undirected graph H is a connected component of the undirected graph G if 

H is a subgraph of G,



H is connected,



No connected subgraph of G has H as a proper subgraph. [Contains maximum number of edges]

THEOREM 2 There is a path between every pair of distinct vertices of a connected undirected graph

PROOF: Direct Proof 1) Let u and v be 2 distinct vertices of G. There is at least one walk between u and v, by definition. 2) Choose the walk of least length. 3) This walk of least length is a path. PROOF: Contrapositive Proof 1) Let 𝑣𝑂 , 𝑣1 … 𝑣𝑛 be the vertex sequence of a walk between distinct vertices v0 and vn. 2) By definition, this walk exists because the graph is connected. If the walk is not a path, then 𝑣𝑖 = 𝑣𝑗 for some i and j with 0 ≤ ⅈ ≤ 𝑗. 3) Therefore, there is a walk from 𝑣0 to 𝜈𝑛 of shorter length with vertex sequence 𝜈0 … 𝜈𝑖−1 𝑣𝑗 … 𝜈𝑛 obtained by deleting the edges corresponding to the vertex sequence 𝜈𝑖 … 𝑣𝑗−1 .

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CS1231 Graph Theory Summary Notes

Section 3 Euler Tour 1. An Euler trail in G is a trail traversing every edge of G. 2. A closed walk is a walk with positive length that starts and ends at the same vertex (i.e. same origin and terminus). 3. A tour in G is a closed walk that traverses each edge if G at least once. 4. An Euler tour in G is a tour traversing each edge exactly once (i.e. closed Euler trail).

THEOREM 3 If a non-empty connected multigraph has an Euler tour, it has no vertices of odd degree.

PROOF: Contrapositive Proof If some vertex of a non-empty connected multigraph has odd degree, it does not have an Euler tour. 1) Let G be the graph with an Euler tour C starting and ending at vertex u. 2) Each time a vertex v occurs as an internal vertex of C, 2 of its edges are accounted for. 3) Since C contains every edge of G, 𝑑𝐺 (𝑣) is even for all 𝑣 ≠ 𝑢. 4) Since C starts and ends at u, 𝑑𝐺 (𝑢) is even.

THEOREM 4 If a non-empty connected multigraph has no vertices of odd degree, it has an Euler tour.

PROOF: Direct Proof 1) Pick any vertex u of graph G to start. 2) Since very vertex has even degree, an edge must be going out. 3) Therefore a closed trail C can be chosen from G. 4) If C contains every edge of G, it is an Euler tour of G. 5) Otherwise, construct subgraph G’ by removing all edges of C from G and any resulting isolated vertices. G’ may be discounted but every vertex of G’ has even degree. 6) Since G is connected, a vertex u’ common to C and G’ can be picked. 7) Starting at u’, choose a closed trail C’ from G’. 8) Patch C and C’ together into one closed trail C’’. 9) Let C = C’’ and go back to STEP 3. YANG LU

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CS1231 Graph Theory Summary Notes PROOF: Contradiction [ALTERNATIVE] 1) Suppose no vertex connected to C and G’ can be produced. 2) No vertex G’ is incident with edge of C. 3) There does not exist a walk between C and G’. 4) A contradiction is derived. 5) Therefore, graph G is no longer connected.

THEOREM 5 A non-empty connected multigraph has an Euler tour if and only if it has no vertices of odd degree.

COROLLARY 3 A non-empty connected multigraph has an Euler trail but not an Euler tour if and only if it has exactly 2 vertices of odd degree.

Section 4 Hamilton Cycles 1. A Hamilton path in G is a path containing every vertex of G. 2. A cycle in G is a closed trail whose origin and internal vertices are distinct. 3. A Hamilton cycle in G is a cycle containing every vertex of G.

PROPOSITION: If a multigraph G has a Hamilton cycle, G has a subgraph H with the following properties: a. 𝑣(𝐻) = 𝑣(𝐺) b. H is connected c. |𝐸(𝐻)| = |𝑉(𝐻)| d. ∀𝑣 ∈ 𝑉(𝐻) 𝑑𝐻 (𝑣) = 2 CONTRAPOSITIVE: If a multigraph G does not have a subgraph H with properties a to b, G does not have a Hamilton cycle.

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CS1231 Graph Theory Summary Notes

Section 5 Graph Representation 1. Let G be an undirected graph and the vertices are ordered as 𝑉(𝐺) = {𝑣1 … 𝑣𝑛 }. The adjacency matrix of G is the 𝑛 × 𝑛 matrix 𝐴(𝐺) = |𝑎𝑖𝑗 | such that 𝑎𝑖𝑗 = |{ ⅇ ∈ 𝐸(𝐺)| 𝑓𝐺 (ⅇ) = {𝑣𝑖 , 𝑣𝑗 }}| Remark: 𝐴(𝐺) is symmetric (i.e. 𝑎𝑖𝑗 = 𝑎𝑗𝑖 for 1 ≤ ⅈ, 𝑗 ≤ 𝑛). 2. Let D be a directed graph and the vertices are ordered as 𝑉(𝐷) = {𝑣1 … 𝑣𝑛 }. The adjacency matrix of D is the 𝑛 × 𝑛 matrix 𝐴(𝐷) = |𝑎𝑖𝑗 | such that 𝑎𝑖𝑗 = |{ ⅇ ∈ 𝐸(𝐷)| 𝑓𝐷 (ⅇ) = {𝑣𝑖 , 𝑣𝑗 }}| Remark: 𝐴(𝐷) may not be symmetric 3. Suppose that 𝐺 = (𝑉, 𝐸) is a simple graph where |𝑉| = 𝑛; Suppose that vertices of G are listed arbitrarily as 𝑣1 , 𝜈2 … 𝑣𝑛 ; The adjacency matrix A (or AG) of G, with respect to this listing of the vertices, is the 𝑛 × 𝑛 zeroone matrix with 1 as its (ⅈ, 𝑗)th entry when vi and vj are adjacent, and 0 as its (ⅈ, 𝑗)th entry when they are not adjacent.

THEOREM 6 Let G be a graph with vertices ordered as 𝑉(𝐺) = {𝑣1 … 𝑣𝑛 } and with adjacency matrix 𝐴(𝐺). Then for each non-negative integer k, number of walks of length k from vi to vj equals to the (i, j)th entry of 𝐴(𝐺)𝑘 .

Remark: 𝐴(𝐺)0 = 𝐼.

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