Question

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

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

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...

I.15: If G is a simple graph with at least two vertices, prove
that G has two vertices of the same degree.
Hint: Let G have n vertices. What are possible
different degree values? Different values if G is connected?

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
G be a simple graph with at least two vertices. Prove that there
are two distinct vertices x, y of G such that deg(x)= deg(y).

Graph Theory, discrete math question:
Let G be a graph with 100 vertices, and chromatic number 99.
Prove a lower bound for the clique number of G. Any lower bound
will do, but try to make it as large as you can.
Please follow this hint my professor gave and show your work,
Thank you!!
Hint: can you prove that the clique number is at least 1? Now
how about 2? Can you prove that the clique number must be...

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

Let G be a simple planar graph with fewer than 12
vertices.
a) Prove that m <=3n-6; b) Prove that G has a vertex of degree
<=4.
Solution: (a) simple --> bdy >=3. So 3m - 3n + 6 = 3f
<= sum(bdy) = 2m --> m - 3n + 6 <=0 --> m <= 3n -
6.
So for part a, how to get bdy >=3 and 2m? I need a
detailed explanation
b) Assume all deg >= 5...

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.)

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