Question

let a be a positive integer. Is it possible to have a graph with degrees 1, 2, 3... 4a + 1 if the graph has 4a + 1 vertices

Answer #1

Let n be a positive integer, and let Hn denote the graph whose
vertex set is the set of all n-tuples with coordinates in {0, 1},
such that vertices u and v are adjacent if and only if they differ
in one position. For example, if n = 3, then (0, 0, 1) and (0, 1,
1) are adjacent, but (0, 0, 0) and (0, 1, 1) are not. Answer the
following with brief justification (formal proofs not
necessary):
a....

Let n be a positive integer and G a simple graph of 4n vertices,
each with degree 2n. Show that G has an Euler circuit. (Hint: Show
that G is connected by assuming otherwise and look at a small
connected component to derive a contradiction.)

Let ? be a graph and let ? be a subgraph of ?. Assume ? contains
at least three vertices.
a) Is it possible for ? to be bipartite and for ? to be a
complete graph?
b) Is it possible for ? to be a complete graph and for ? to be
bipartite?

Prove that every graph has two vertices with the same degree.
(hint: what are the possible degrees?)

Let m be a composite positive integer and suppose that m = 4k
+ 3 for some integer k. If m = ab for some integers a and b, then a
= 4l + 3 for some integer l or b = 4l + 3 for some integer l.
1. Write the set up for a proof by contradiction.
2. Write out a careful proof of the assertion by the method of
contradiction.

Let p be an odd prime and let a be an odd integer with p not
divisible by a. Suppose that p = 4a + n2 for some
integer n. Prove that the Legendre symbol (a/p) equals 1.

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
n be a positive integer and let S be a subset of n+1 elements of
the set {1,2,3,...,2n}.Show that
(a) There exist two elements of S that are relatively prime,
and
(b) There exist two elements of S, one of which divides the
other.

Let λ be a positive irrational real number. If n is a positive
integer, choose by the Archimedean Property an integer k such that
kλ ≤ n < (k + 1)λ. Let φ(n) = n − kλ. Prove that the set of all
φ(n), n > 0, is dense in the interval [0, λ]. (Hint: Examine the
proof of the density of the rationals in the reals.)

Very confused about where to place vertices in the graph
according to their the degrees so I can find the edge set...
Construct a simple graph with vertices
O,P,Q,R,S,T
whose degrees are 4, 3, 4, 4, 1, 4
What is the edge set?

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