Let G be a graph in which there is a cycle C odd length that has vertices on all of the other odd...
Let G be a graph in which there is a cycle C odd length that has vertices on all of the other odd cycles. Prove that the chromatic number of G is less than or equal to 5.
Homework Answers
Solution:
Let C be the smallest odd cycle in G. Note that C has no chords (i.e. edges between its vertices that aren't part of the cycle), as otherwise there would be a smaller odd cycle. Hence we can color C with 3 colors, and it will be a valid partial coloring of G.
Now if we throw all the vertices in C from G, the remaining graph is bipartite because any odd cycle in G must share a vertex with C by assumption. Now we can color all of those vertices with two new colors. We have colored G with 55 colors, so χ(G)≤5
Add Answer to:
Let G be a graph in which there is a cycle C odd length that has vertices on all of the other odd...
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2. Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
Prove that an undirected graph is bipartite iff it contains no cycle whose length is odd...
Prove that an undirected graph is bipartite iff it contains no cycle whose length is odd (called simply an "odd cycle"). An undirected graph G = (V,E) is called "bipartite" when the vertices can be partitioned into two subsets V = V_1 u V_2 (with V_1 n V_2 = {}) such that every edge of G has one endpoint in V_1 and the other in V_2 (equivalently, no edge of G has both endpoints in V_1 or both endpoints in...
(5) Let G be a graph without loops. Let n be the number of vertices and...
(5) Let G be a graph without loops. Let n be the number of vertices and let xG(z) be its chromatic polynomial. Recall from HW5 that 2(G) is the number of cycle-free orientations of G. Show that
G is a bipartite graph. Show G doesn't have an odd cycle. (Def. of odd cycle:...
G is a bipartite graph. Show G doesn't have an odd cycle. (Def. of odd cycle: simple cycle w/ odd number of vertices)
Let G be a graph with n vertices and n edges. (a) Show that G has...
Let G be a graph with n vertices and n edges. (a) Show that G has a cycle. (b) Use part (a) to prove that if G has n vertices, k components, and n − k + 1 edges, then G has a cycle.
A random undirected graph has 9 vertices. An unordered cycle is a connection within the graph...
A random undirected graph has 9 vertices. An unordered cycle is a connection within the graph that connects a number of vertices. For example an unordered cycle of 3 would be a triangle within the graph of 3 connected vertices. To find the total number of possible unordered cycles of 3 vertices from a total of 9 you can use the Combination Formula C(n,r) = n!/r!(n-r)! which is total number of possible combinations of r objects from a set of...
Given a graph G with no odd cycles, we want to color the vertices with two...
Given a graph G with no odd cycles, we want to color the vertices with two colors, red and white (go Huskers!) such that no two neighbors have the same color. Give a linear algorithm to accomplish this, analyze its running time, and prove it is correct.
7.5 (i) Prove that, if G is a bipartite graph with an odd number of vertices,...
7.5 (i) Prove that, if G is a bipartite graph with an odd number of vertices, then G is non-Hamiltonian. (ii) Deduce that the graph in Fig. 7.7 is non-Hamiltonian. Fig. 7.7 (iii) Show that, if n is odd, it is not possible for a knight to visit all the squares of an n chessboard exactly once by knight's moves and return to its starting point.
Let n >k> 1 with n even and k odd. Make a k-regular graph G by putting n vertices in a circle and...
need help with a and b in this graph theory question
Let n >k> 1 with n even and k odd. Make a k-regular graph G by putting n vertices in a circle and connecting each vertex to the exact a) Show that for all u,v there are k internally disjoint u, v-paths (you (b) Use the previous part, even if you did not prove it, to show that the e vertex and the k 1 closest vertices on either...
Problem 12.29. A basic example of a simple graph with chromatic number n is the complete graph on...
Problem 12.29. A basic example of a simple graph with chromatic number n is the complete graph on n vertices, that is x(Kn) n. This implies that any graph with Kn as a subgraph must have chromatic number at least n. It's a common misconception to think that, conversely, graphs with high chromatic number must contain a large complete sub- graph. In this problem we exhibit a simple example countering this misconception, namely a graph with chromatic number four that...
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
(5) Let G be a graph without loops. Let n be the number of vertices and let xG(z) be its chromatic polynomial. Recall from HW5 that 2(G) is the number of cycle-free orientations of G. Show that
Given a graph G with no odd cycles, we want to color the vertices with two colors, red and white (go Huskers!) such that no two neighbors have the same color. Give a linear algorithm to accomplish this, analyze its running time, and prove it is correct.
7.5 (i) Prove that, if G is a bipartite graph with an odd number of vertices, then G is non-Hamiltonian. (ii) Deduce that the graph in Fig. 7.7 is non-Hamiltonian. Fig. 7.7 (iii) Show that, if n is odd, it is not possible for a knight to visit all the squares of an n chessboard exactly once by knight's moves and return to its starting point.
need help with a and b in this graph theory question
Let n >k> 1 with n even and k odd. Make a k-regular graph G by putting n vertices in a circle and connecting each vertex to the exact a) Show that for all u,v there are k internally disjoint u, v-paths (you (b) Use the previous part, even if you did not prove it, to show that the e vertex and the k 1 closest vertices on either...
Problem 12.29. A basic example of a simple graph with chromatic number n is the complete graph on n vertices, that is x(Kn) n. This implies that any graph with Kn as a subgraph must have chromatic number at least n. It's a common misconception to think that, conversely, graphs with high chromatic number must contain a large complete sub- graph. In this problem we exhibit a simple example countering this misconception, namely a graph with chromatic number four that...
Most questions answered within 3 hours.
-
Where is the error in this code sequence?
String s1 = "Hello";
String s2 = "ello";...
asked 11 months ago -
Financial data for Joel de Paris, Inc., for last year
follow:
Joel de Paris, Inc.
Balance...
asked 11 months ago -
Consider this reaction:
Al2(SO4)3 (aq)+ BaCl3
(aq) Al2Cl6 (aq)- +
3BaSO4(s) . What is the...
asked 11 months ago -
Suppose that Savneet is considering increasing her
recent random sample from 20 car rentals to 40...
asked 11 months ago -
Trucks arrive at an unloading terminal at an average rate of 120
per hour.
Trucks arrive...
asked 11 months ago -
Why are methanol and ethanol completely soluble in water while
octanol is not very little soluble....
asked 11 months ago -
A facilities manager at a university reads in a research report
that the mean amount of...
asked 11 months ago -
When the CuSO4 is rehydrated by adding water to the anhydrous
compound, is this an endothermic...
asked 11 months ago -
A ray of sunlight is passing from diamond into crown glass; the
angle of incidence is...
asked 11 months ago -
A block of mass 0.249 kg is placed on top of a light, vertical
spring of...
asked 11 months ago -
how do the kidneys compensate in the presences of acidosis
a) trigger hyperventilate
b) reserve acid...
asked 11 months ago -
Question 501 pts
The rental rate of capital to the firm increases. Which of the
following...
asked 11 months ago