Question

prove that every graph has an even number of odd nodes

Answer #1

Prove that a full non-empty binary tree must have an odd number
of nodes via induction

Let G be a graph where every vertex has odd degree, and G has a
perfect matching. Prove that if M is a perfect matching of G, then
every bridge of G is in M.
The Proof for this question already on Chegg is wrong

How would you prove that for every natural number n, the product
of any n odd numbers is odd, using mathematical induction?

prove that the sum of two odd integers is even

Perform the following tasks:
a. Prove directly that the product of an even and an odd number
is even.
b. Prove by contraposition for arbitrary x does not equal -2: if
x is irrational, then so is x/(x+2)
c. Disprove: If x is irrational and y is irrational, then x+y is
irrational.

A second order homogeneous linear differential equation has
odd-even parity. Prove that if one of its solutions is an even
function, the other can be constructed as an odd function.

The distance between two connected nodes in a graph is the
length (number of edges) of the shortest path connecting them. The
diameter of a connected graph is the maximum distance between any
two of its nodes. Let v be an arbitrary vertex in a graph G. If
every vertex is within distance d of v, then show that the diameter
of the graph is at most 2d.

Theorem: If m is an even number and n is an odd number, then
m^2+n^2+1 is even. Don’t prove it.
In writing a proof by contraposition, what is your “Given”
(assumption)? ___________________________
What is “To Prove”: _____________________________

Prove that if G is a graph in which any two nodes are connected
by a unique path, then G is a tree.

Let G be a graph in which there is a cycle C odd length that has
vertices on all of the other odd cycles. Prove that the chromatic
number of G is less than or equal to 5.

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