Let G be a simple graph with at least four vertices. a) Give an example to show that G can contai...

Question

Question

Let G be a simple graph with at least four vertices.

a) Give an example to show that G can contain a closed Eulerian trail, but not a Hamiltonian cycle.

b) Give an example to show that G can contain a closed Hamiltonian cycle, but not a Eulerian trail.

Answer 1

Answer #1

Let ba simple raph toith attost e Eanbl to shes th an ntain a closnd e ulenan thain a cedcba Rdt thete ho Hami ltonian (ycle

Answer 2

Similar Homework Help Questions

A connected simple graph G has 16 vertices and 117 edges. Prove G is Hamiltonian and prove G is not Eulerian

A connected simple graph G has 16 vertices and 117 edges. Prove G is Hamiltonian and prove G is not Eulerian
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 simple graph with 2n, n 2 vertices. Suppose there are at least n2 1 edges. Show that at least one triangle i...

Let G be a simple graph with 2n, n 2 vertices. Suppose there are at least n2 1 edges. Show that at least one triangle is formed. Hint: Check n 2 first and then use induction Let G be a simple graph with 2n, n 2 vertices. Suppose there are at least n2 1 edges. Show that at least one triangle is formed. Hint: Check n 2 first and then use induction

(a) Let L be a minimum edge-cut in a connected graph G with at least two vertices. Prove that G −...

(a) Let L be a minimum edge-cut in a connected graph G with at least two vertices. Prove that G − L has exactly two components. (b) Let G an eulerian graph. Prove that λ(G) is even.
* Exercise 1: Let G be the graph with vertex set V(G) = Zi,-{0,-, that two...

* Exercise 1: Let G be the graph with vertex set V(G) = Zi,-{0,-, that two vertices x, y E V(G) are connected by an edge if and only if ,10) and such ryt5 mod 11 or xEy t7 mod 11 1. Draw the graph G. 2. Show that the graph G is Eulerian, i.e., it has a closed trail containing all its edges
Give an example of a graph G with at least 10 vertices such that the greedy...

Give an example of a graph G with at least 10 vertices such that the greedy 2-approximation algorithm for Vertex-Cover given below is guaranteed to produce a suboptimal vertex cover. Algorithm Vertex CoverApprox(G): Input: A graph G Output: A small vertex cover C for G while G still has edges do select an edge e (v, w) of G add vertices v and w to C for each edge f incident to v or w do remove f from G...

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.
G3: I can determine whether a graph has an Euler trail (or circuit), or a Hamiltonian...

G3: I can determine whether a graph has an Euler trail (or circuit), or a Hamiltonian path (or cycle), and I can clearly explain my reasoning. Answer each question in the space provided below. 1. Draw a simple graph with 7 vertices and 11 edges that has an Euler circuit. Demonstrate the Euler circuit by listing in order the vertices on it. 2. For what pairs (m, n) does the complete bipartite graph, Km,n contain a Hamiltonian cycle? Justify your...
1- Give an example (by drawing or by describing) of the following undirected graphs (a) A...

1- Give an example (by drawing or by describing) of the following undirected graphs (a) A graph where the degree in each vertex is even and the total number of edges is odd (b) A graph that does not have an eulerian cycle. An eulerian cycle is a cycle where every edge of the graph is visited exactly once. (c) A graph that does not have any cycles and the maximum degree of a node is 2 (minimum degree can...

Let G be a graph with n vertices. Show that if the sum of degrees of...

Let G be a graph with n vertices. Show that if the sum of degrees of every pair of vertices in G is at least n − 1 then G is connected.