Question 1
a) Prove that if u and v are distinct vertices of a graph G,
there exists a walk from u to v if and only if there exists a path
(a walk with distinct vertices) from u to v.
b) Prove that a graph is bipartite if and only if it contains no
cycles of odd length.
Please write legibly with step by step details. Many thanks!
Prove
1. For each u ∈ R n there is a v ∈ R n such that u + v= 0
2. For all u, v ∈ R n and a ∈ R, a(u + v) = au + av
3. For all u ∈ R n and a, b ∈ R, (a + b)u = au + bu
4. For all u ∈ R n , 1u=u
1. Prove that for any graph, the sum the degreesPv∈V
deg(v) is twice the number of edges |E|. (By “prove” I mean write a
few sentences explaining why it is true.)
2. i) At a recent math seminar, 5 mathematicians greeted each
other by shaking hands. Is it possible for each mathematician to
shake hands with exactly 3 other people? (No one can shake his or
her own hand.) To answer the question, please rephrase the problem
as a problem about...
Assume that G is connected and let
(G) = min{deg(v) ∈ Z ≥0 | v ∈ V (G)}
denote the lowest vertex degree that occurs in a graph G and
let
λ(G) = min{|K| ∈ Z ≥0 | K ⊆ E(G) is a cutset of G}
denote the edge connectivity of G.
Prove that λ(G) ≤ δ(G)
A maximal plane graph is a plane graph G = (V, E) with n ≥ 3
vertices such that if we join any two non-adjacent vertices in G,
we obtain a non-plane graph.
a) Draw a maximal plane graphs on six vertices.
b) Show that a maximal plane graph on n points has 3n − 6 edges
and 2n − 4 faces.
c) A triangulation of an n-gon is a plane graph whose infinite
face boundary is a convex n-gon...
Let u and v be two integers and let us assume u^2 + uv +v^2 is
divisible by 9. Show that then u and v are divisible by 3. (please
do this by contrapositive).