Question 1

ﬁll in the blanks.
-The largest value of n for which $K _ { n }$ is planar is ____.

Accepted Answer

The answer of ﬁll in the blanks.
-The largest value of...

Question 2

ﬁll in the blanks.
-List all positive integers n such that $C _ { n }$ is bipartite ____.

Accepted Answer

The answer of ﬁll in the blanks.
-List all positive integers...

Question 3

ﬁll in the blanks.
-List all positive integers n such that $W _ { n }$ has an Euler circuit ____.

Accepted Answer

The answer of ﬁll in the blanks.
-List all positive integers...

Question 4

either give an example or prove that there are none.
-A simple graph with 6 vertices and 16 edges.

Accepted Answer

None. The largest nu

Question 5

the grid graph GS1U1B1m,nS1U1B0 refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example:  
-For which positive integers m and n does $G _ { m , n }$ have an Euler circuit?

Accepted Answer

The answer of the grid graph GS1U1B1m,nS1U1B0 refers to the...

Question 6

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph.
-$K _ { 6 }$

Accepted Answer

\[\text { Vertices } = \{ 1,2,3,4,5,6 \}

Question 7

ﬁll in the blanks.
-There are ____ non-isomorphic simple graphs with 3 vertices.

Accepted Answer

The answer of ﬁll in the blanks.
-There are ____ non-isomorphic...

Question 8

The picture at the right shows the ﬂoor plan of an oﬃce. once, starting and ending at A. plan a walk that passes through each doorway exactly Use graph theory ideas to prove that it is impossible to plan a walk that passes through each doorway exactlyonce, starting and ending at A.

Accepted Answer

Use vertices for rooms and edg

Question 9

either give an example or prove that there are none.
-A simple graph with degrees 1, 1, 2, 4.

Accepted Answer

None. In a simple gr

Question 10

ﬁll in the blanks.
-$W _ { n }$ has ____ edges and ____ vertices.

Accepted Answer

The answer of ﬁll in the blanks.
-\(W _ { n...

Question 11

either give an example or prove that there are none.
-A simple digraph with indegrees 0, 1, 2, 4, 5 and outdegrees 0, 3, 3, 3, 3.

Accepted Answer

None. In a simple gr

Question 12

ﬁll in the blanks.
-The length of the longest simple circuit in $K _ { 5 }$ is ____.

Accepted Answer

The answer of ﬁll in the blanks.
-The length of the...

Question 13

the grid graph GS1U1B1m,nS1U1B0 refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example:  
-Find the region-chromatic number for Gm,n (including the inﬁnite face).

Accepted Answer

The answer of the grid graph GS1U1B1m,nS1U1B0 refers to the...

Question 14

ﬁll in the blanks.
-There are ____ 0's and ____ 1's in the adjacency matrix for $C _ { n }$

Accepted Answer

The answer of ﬁll in the blanks.
-There are ____ 0's...

Question 15

ﬁll in the blanks.
-List all positive integers n such that $K _ { n }$ has an Euler circuit ____.

Accepted Answer

The answer of ﬁll in the blanks.
-List all positive integers...

Question 16

A simple graph is regular if every vertex has the same degree. 
(a) For which positive integers n are the following graphs regular: Cn, Wn, Kn, Qn? 
(b) For which positive integers m and n is Km,n regular?

Accepted Answer

(a) All @#LAT-DLM&,

Question 17

Use Dijkstra's Algorithm to ﬁnd the shortest path length between the vertices a and z in this weighted graph.

Accepted Answer

First iteration: distinguished vertices

Question 18

ﬁll in the blanks.
-Every Euler circuit for $K _ { 9 }$ has length ____ .

Accepted Answer

The answer of ﬁll in the blanks.
-Every Euler circuit for...

Question 19

either give an example or prove that there are none.
-A graph with a Hamilton path but no Hamilton circuit.

Accepted Answer

The answer of either give an example or prove that...

Question 20

ﬁll in the blanks.
-List all positive integers n such that $W _ { n }$ has a Hamilton circuit ____.

Accepted Answer

The answer of ﬁll in the blanks.
-List all positive integers...

Exam 10: A: Graphs

ﬁll in the blanks. -The largest value of n for which KnK _ { n }Kn​ is planar is ____.

ﬁll in the blanks. -List all positive integers n such that CnC _ { n }Cn​ is bipartite ____.

ﬁll in the blanks. -List all positive integers n such that WnW _ { n }Wn​ has an Euler circuit ____.

either give an example or prove that there are none. -A simple graph with 6 vertices and 16 edges.

the grid graph Gm,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -For which positive integers m and n does Gm,nG _ { m , n }Gm,n​ have an Euler circuit?

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. - K6K _ { 6 }K6​

ﬁll in the blanks. -There are ____ non-isomorphic simple graphs with 3 vertices.

The picture at the right shows the ﬂoor plan of an oﬃce. once, starting and ending at A. plan a walk that passes through each doorway exactly Use graph theory ideas to prove that it is impossible to plan a walk that passes through each doorway exactlyonce, starting and ending at A.

either give an example or prove that there are none. -A simple graph with degrees 1, 1, 2, 4.

ﬁll in the blanks. - WnW _ { n }Wn​ has ____ edges and ____ vertices.

either give an example or prove that there are none. -A simple digraph with indegrees 0, 1, 2, 4, 5 and outdegrees 0, 3, 3, 3, 3.

ﬁll in the blanks. -The length of the longest simple circuit in K5K _ { 5 }K5​ is ____.

the grid graph Gm,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -Find the region-chromatic number for Gm,n (including the inﬁnite face).

ﬁll in the blanks. -There are ____ 0's and ____ 1's in the adjacency matrix for CnC _ { n }Cn​

ﬁll in the blanks. -List all positive integers n such that KnK _ { n }Kn​ has an Euler circuit ____.

A simple graph is regular if every vertex has the same degree. (a) For which positive integers n are the following graphs regular: Cn, Wn, Kn, Qn? (b) For which positive integers m and n is Km,n regular?

Use Dijkstra's Algorithm to ﬁnd the shortest path length between the vertices a and z in this weighted graph.

ﬁll in the blanks. -Every Euler circuit for K9K _ { 9 }K9​ has length ____ .

either give an example or prove that there are none. -A graph with a Hamilton path but no Hamilton circuit.

ﬁll in the blanks. -List all positive integers n such that WnW _ { n }Wn​ has a Hamilton circuit ____.

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

ﬁll in the blanks. -The largest value of n for which $K _ { n }$ is planar is ____.

ﬁll in the blanks. -List all positive integers n such that $C _ { n }$ is bipartite ____.

ﬁll in the blanks. -List all positive integers n such that $W _ { n }$ has an Euler circuit ____.

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. - $K _ { 6 }$

ﬁll in the blanks. - $W _ { n }$ has edges and vertices.

ﬁll in the blanks. -The length of the longest simple circuit in $K _ { 5 }$ is ____.

the grid graph G_m,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -Find the region-chromatic number for Gm,n (including the inﬁnite face).

ﬁll in the blanks. -There are 0's and 1's in the adjacency matrix for $C _ { n }$

ﬁll in the blanks. -List all positive integers n such that $K _ { n }$ has an Euler circuit ____.

A simple graph is regular if every vertex has the same degree. (a) For which positive integers n are the following graphs regular: C_n, W_n, K_n, Q_n? (b) For which positive integers m and n is K_m,n regular?

ﬁll in the blanks. -Every Euler circuit for $K _ { 9 }$ has length ____ .

ﬁll in the blanks. -List all positive integers n such that $W _ { n }$ has a Hamilton circuit ____.