Question

show that any simple, connected graph with 31 edges and 12 vertices
is not planar.

Answer #1

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30. a) Show if G is a connected planar simple graph with v
vertices and e edges with v ≥ 3 then e ≤ 3v−6.
b) Further show if G has no circuits of length 3 then e ≤
2v−4.

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.

A graph is called planar if it can be
drawn in the plane without any edges crossing. The Euler’s formula
states that v − e + r = 2,
where v,e, and r are the numbers of vertices,
edges, and regions in a planar graph, respectively. For the
following problems, let G be a planar simple graph with 8
vertices.
Find the maximum number of edges in G.
Find the maximum number of edges in G, if G
has no...

Let there be planar graph G with 12 vertices where every
vertices may or may not be connected by an edge. The edges in G
cannot intersect. What is the maximum number of edges in G. Draw an
example of G. What do you notice about the faces and the maximum
number of edges?

Draw an example of a connected bipartite simple graph with 9
vertices and 10 edges that has an Euler tour.

Suppose that a connected planar graph has eight vertices each of
degree 3 then how many regions does it have?And suppose that a
polyhedron has 12 triangular faces then determine the number of
edges and vertices.

Prove that every connected planar graph with less than 12
vertices can be 4-colored

Use proof by induction to prove that every connected planar
graph with less than 12 vertices has a vertex of degree at most
4.

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 connected planar graph with 3 or more vertices which
is drawn in the plane. Let ν, ε, and f be as usual. a) Use P i fi =
2ε to show that f ≤ 2ε 3 . b) Prove that ε ≤ 3ν − 6. c) Use b) to
show that K5 is not planar.

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