Question

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).

Answer #1

I.15: If G is a simple graph with at least two vertices, prove
that G has two vertices of the same degree.
Hint: Let G have n vertices. What are possible
different degree values? Different values if G is connected?

Let u and v be distinct vertices in a graph G. Prove that there
is a walk from ? to ? if and only if there is a path from ? to
?.

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...

Let G be a simple graph with n(G) > 2. Prove that G is
2-connected iff for every set of 3 distinct vertices, a,
b and c, there is an a,c-path
that contains b.

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

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.

(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.

Let G = (X, E) be a connected graph. The distance between two
vertices x and y of G is the shortest length of the paths linking x
and y. This distance is denoted by d(x, y). We call the center of
the graph any vertex x such that the quantity max y∈X d(x, y) is
the smallest possible. Show that if G is a tree then G has either
one center or two centers which are then neighbors

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.

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.

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