Let's say that m is the number of edges in an undirected graph. 1. Show that...
Let's say that m is the number of edges in an undirected graph.
1. Show that if the degree of every vertex is at least k, then the graph has a simple path of length at least k.
2. Show that every graph has a subgraph with minimum degree m/n. Hint: iteratively remove all vertex of degree strictly smaller than m/n.
3. Show that any graph has a path of length at least m/n. Use the two claims proven in the previous questions.
Homework Answers
Add Answer to:
Let's say that m is the number of edges in an undirected
graph.
1. Show that...
Long paths in undirected graphs In this question m is the number of edges in an...
Long paths in undirected graphs In this question m is the number of edges in an undirected graph. 1. Show that if the degree of every vertex is at least k, then the graph has a simple path of length at least k. Hint: consider the longest simple path in the graph say from x to y. Show that the endpoints x and y do not have edges to vertices outside the path. Thus all the neighbors of x, y...
Bounds on the number of edges in a graph. (a) Let G be an undirected graph...
Bounds on the number of edges in a graph. (a) Let G be an undirected graph with n vertices. Let Δ(G) be the maximum degree of any vertex in G, δ(G) be the minimum degree of any vertex in G, and m be the number of edges in G. Prove that δ(G)n2≤m≤Δ(G)n2
An undirected bipartite graph has n vertices and m edges. a) If the graph is connected,...
An undirected bipartite graph has n vertices and m edges. a) If the graph is connected, what is the minimum number of edges? b) If the graph is disconnected, what is the maximum number of edges? c) What is the longest single path? d) If the path can pass through a vertex and not any edges more than once, What is the longest path? Kindly provide me with an example for me to relate
Question 1: Given an undirected connected graph so that every edge belongs to at least one...
Question 1: Given an undirected connected graph so that every edge belongs to at least one simple cycle (a cycle is simple if be vertex appears more than once). Show that we can give a direction to every edge so that the graph will be strongly connected. Question 2: Given a graph G(V, E) a set I is an independent set if for every uv el, u #v, uv & E. A Matching is a collection of edges {ei} so...
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that ther...
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
A 2-coloring of an undirected graph with n vertices and m edges is the assignment of one of two colors (say, red or green) to each vertex of the graph
A 2-coloring of an undirected graph with n vertices and m edges is the assignment of one of two colors (say, red or green) to each vertex of the graph, so that no two adjacent nodes have the same color. So, if there is an edge (u,v) in the graph, either node u is red and v is green or vice versa. Give an O(n + m) time algorithm (pseudocode!) to 2-colour a graph or determine that no such coloring...
Write down true (T) or false (F) for each statement. Statements are shown below If a...
Write down true (T) or false (F) for each statement. Statements are shown below If a graph with n vertices is connected, then it must have at least n − 1 edges. If a graph with n vertices has at least n − 1 edges, then it must be connected. If a simple undirected graph with n vertices has at least n edges, then it must contain a cycle. If a graph with n vertices contain a cycle, then it...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more tha...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....
Say that we have an undirected graph G(V, E) and a pair of vertices s, t and a vertex v that we call a a desired middle vertex . We wish to find out if there exists a simple path (every vertex appears...
Say that we have an undirected graph G(V, E) and a pair of vertices s, t and a vertex v that we call a a desired middle vertex . We wish to find out if there exists a simple path (every vertex appears at most once) from s to t that goes via v. Create a flow network by making v a source. Add a new vertex Z as a sink. Join s, t with two directed edges of capacity...
Exercise 5. Let G be a graph in which every vertex has degree at least m....
Exercise 5. Let G be a graph in which every vertex has degree at least m. Prove that there is a simple path (i.e. no repeated vertices) in G of length m.
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....
Exercise 5. Let G be a graph in which every vertex has degree at least m. Prove that there is a simple path (i.e. no repeated vertices) in G of length m.
Most questions answered within 3 hours.
-
Do
you think that apps for online shopping will take over websites
?
asked 22 seconds ago -
you’ll write a program to illustrate how Linked list and
structure are used in practice, we’ll...
asked 35 seconds ago -
Suppose that each student has probability p of correctly
answering a question chosen at random from...
asked 9 minutes ago -
Find the critical values: A. z-value a= 0.12 and left-tailed.
B.) z-value a=0.08 and two-tailed. C.)...
asked 16 minutes ago -
$2,300 down payment, $13,700, 48-month loan at 6 percent
interest requiring a monthly payment of $321.74;...
asked 12 minutes ago -
College students average 9 hours of sleep per night with a
standard deviation of 45 minutes....
asked 38 minutes ago -
Charge q1= 2.0nC is located at (-1.0 cm, 0), charge q2= -2.0nC
is located at (1...
asked 45 minutes ago -
problem 17.46
In a laboratory experiment, one end of a horizontal string is
tied to a...
asked 43 minutes ago -
How do you know if you have all the necessary information to
make a forecast? Isn't...
asked 45 minutes ago -
A person perspires. How much liquid water (as a percentage of
the mass of the person)...
asked 44 minutes ago -
a: A car is accelerating uniformly from a rest state at 4 m/s^2.
What distance does...
asked 49 minutes ago -
How many integers between 1 and 943 inclusive are divisible by 5
or 3?
asked 1 hour ago