Question

A K-regular graph G is a graph such that deg(v) = K for all vertices v in G. For example, c_9 is a 2-regular graph, because every vertex has degree 2. For some K greater than or equal to 2, neatly draw a simple K-regular graph that has a bridge. If it is impossible, prove why.

Answer #1

a
graph is regular of degree k if every vertex has the same degree,
k. show that G has a hamiltonian circuit if G has 13 vertices and
is regular of degree 6.

a. A graph is called k-regular if every vertex has degree k.
Explain why any k-regular graph with 11 vertices must conain an
Euler circuit. (Hint – think of the possible values of k). b. If G
is a 6-regular graph, what is the chromatic number of G? Must it be
6? Explain

Graph Theory.
A simple graph G with 7 vertices and 10 edges has the
following properties: G has six vertices of degree
a and one vertex of degree b. Find a and
b, and draw the graph.
Show all work.

Suppose we are going to color the vertices of a connected planar
simple graph such that no two adjacent vertices are with the same
color.
(a) Prove that if G is a connected planar simple graph, then G
has a vertex of degree at most five.
(b) Prove that every connected planar simple graph can be
colored using six or fewer colors.

Let G be a simple planar graph with fewer than 12
vertices.
a) Prove that m <=3n-6; b) Prove that G has a vertex of degree
<=4.
Solution: (a) simple --> bdy >=3. So 3m - 3n + 6 = 3f
<= sum(bdy) = 2m --> m - 3n + 6 <=0 --> m <= 3n -
6.
So for part a, how to get bdy >=3 and 2m? I need a
detailed explanation
b) Assume all deg >= 5...

Q. a graph is called k-planar if every vertex has degree
k.
(a) Explain why any k-regular graph with 11 vertices should contain
an Euler’s circuit.
(what are the possible values of k).
(b) Suppose G is a 6-regular graph. Must the chromatic number of G
be at leat 6? Explain.

Draw an undirected graph with 6 vertices that has an Eulerian
Cycle and a Hamiltonian Cycle. The degree of each vertex
must be greater than 2. List the degrees of the
vertices, draw the Hamiltonian Cycle on the graph and give the
vertex list of the Eulerian Cycle.
Draw a Bipartite Graph with 10 vertices that has an Eulerian
Path and a Hamiltonian Cycle. The degree of each vertex
must be greater than 2. List the degrees of the
vertices, draw the Hamiltonian Cycle...

Let G be a simple graph in which all vertices have degree four.
Prove that it is possible to color the edges of G orange or blue so
that each vertex is adjacent to two orange edges and two blue
edges.
Hint: The graph G has a closed Eulerian walk. Walk along it and
color the edges alternately orange and blue.

Let
G be a simple graph with at least two vertices. Prove that there
are two distinct vertices x, y of G such that deg(x)= deg(y).

Graph Theory
Prove that if G is a graph with x(G-v-w)=x(G)-2 for every pair
of vertices v and w in G, then G is complete.
Hint: assume G is not complete.

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