Question

For what values of m, n ∈ Z>0 does the complete bipartite graph Km,n have a perfect matching? Prove it

Answer #1

Prove that the complete graph K2n has a perfect
matching for all n ∈ Z>0

Find the diameters of Kn (Connected graph with n vertices), Km,n
(Bipartite graph with m and n vertices), and Cn (Cycle graph with n
vertices). For each, clearly explain your reasoning.

Prove that a bipartite simple graph with n vertices must have at
most n2/4 edges. (Here’s a hint. A bipartite graph would have to be
contained in Kx,n−x, for some x.)

For which values of n ≥ 3 do these graphs have an Euler
cycle?
(a) Complete graph Kn
(b) Cycle graph Cn
(c) Complete bipartite graph Kn,n

Graph Theory:
If a connedted bipartite graph has a Hamilton path, then m and
n can differ at most one.
why is that? shouldn't there be an even amount of
vertices?
please explain

G is a complete bipartite graph on 7 vertices.
G is planar, and it has an Eulerian path. Answer the questions, and
explain your answers.
1. How many edges does G have?
2. How many faces does G have?
3. What is the chromatic number of G?

Consider the complete bipartite graph Kn,n with 2n vertices. Let
kn be the number of edges in Kn,n. Draw K1,1, K2,2 and K3,3 and
determine k1, k2, k3. Give a recurrence relation for kn and solve
it using an initial value.

Draw an Euler graph on an even number of vertices, that does not
have perfect matching. Prove it.

How many vertices and edges does the complete tripartite graph
K_m,n,p have? Prove your conjecture.

What does it mean that the
graph starts at 0 km/s and then quickly climbs to >200 km/s
before flattening out?

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