Question

a) Let k>1 be the size of a minimum edge cut in G. Show that the deletion of k edges from G results in at most 2 components.

b) Is the same true for vertex cuts? Justify your answer.

Answer #1

(a) Let L be a minimum edge-cut in a connected graph G with at
least two vertices. Prove that G − L has exactly two
components.
(b) Let G an eulerian graph. Prove that λ(G) is even.

Let T be a minimum spanning tree of graph G obtained by Prim’s
algorithm. Let Gnew be a graph obtained by adding to G a new vertex
and some edges, with weights, connecting the new vertex to some
vertices in G. Can we construct a minimum spanning tree of Gnew by
adding one of the new edges to T ? If you answer yes, explain how;
if you answer no, explain why not.

Proof: Let G be a k-connected k-regular graph. Show that, for
any edge e, G has a perfect matching M such that e ε
M.
Please show full detailed proof. Thank you in advance!

A graph G is said to be k-critical if ?(?)=? and the deletion of
any vertex yields a graph of smaller chromatic number.
(i) Find all 2-critical and 3-critical simple graphs. Be sure to
justify your answer.

Let G be a simple graph having at least one edge, and let L(G)
be its line graph. (a) Show that χ0(G) = χ(L(G)). (b) Assume that
the highest vertex degree in G is 3. Using the above, show Vizing’s
Theorem for G. You may use any theorem from class involving the
chromatic number, but no theorem involving the chromatic index

Show Proof of correctness and state, and solve the Recurrence
using the Master Theorem. Let G = G(V, E) be an arbitrary,
connected, undirected graph with vertex set V and edge set E.
Assume that every edge in E represents either a road or a bridge.
Give an efficient algorithm that takes as input G and decides
whether there exists a spanning tree of G where the number of edges
that represents roads is
floor[ (|V|/ √ 2) ]. Do...

Show that if G is connected with n ≥ 2 vertices and n − 1 edges
that G contains a vertex of degree 1.
Hint: use the fact that deg(v1) + ... + deg(vn) = 2e

Let G be the graph obtained by erasing one edge from K5. What is
the chromatic number of G? Prove your answer.

Let G be an n-vertex graph with n ≥ 2 and δ(G) ≥ (n-1)/2. Prove
that G is connected and that the diameter of G is at most two.

Let G be a group with subgroups H and K.
(a) Prove that H ∩ K must be a subgroup of G.
(b) Give an example to show that H ∪ K is not necessarily a
subgroup of G.
Note: Your answer to part (a) should be a general proof that the
set H ∩ K is closed under the operation of G, includes the identity
element of G, and contains the inverse in G of each of its
elements,...

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