Question

Graph theory graph colouring

If H is a subgraph of G, then show that χ(H) ≤ χ(G).

Answer #1

Use induction to prove that every graph G = (V, E) satisfies
χ(G) ≤ ∆(G).

Let G be a graph on 10 vertices that has no triangles. Show that
Gc (G complement) must have K4 subgraph

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...

A spanning tree of connected graph G = (V, E) is an acyclic
connected subgraph (V, E0 ) with the same vertices as G. Show that
every connected graph G = (V, E) contains a spanning tree. (It is
the connected subgraph (V, E0 ) with the smallest number of
edges.)

Prove that the number χ(G, n) of valid n-colorings of a
multigraphs satisfies the formula χ(G, n) = χ(G − e, n) − χ(G/e,
n). Explain the meaning of this formula when there are several
edges connecting the endpoints of the edge e.

Graph Theory.
A simple graph G with 7 vertices and 10 edges has the
following properties: G has six vertices of degree
a and one vertex of degree b. Find a and
b, and draw the graph.
Show all work.

Intro to graph theory question:
1) Draw a graph G with w(G) = 2 (w(g) is clique number)
and x(G) = 5 (x(g) is chromatic number)

graph theory
give an example or show that no such example exists of a
nonregular graph whose complement is regular.

Graph Theory
Let v be a vertex of a non trivial graph G. prove that if G is
connected, then v has a neighbor in every component of G-v.

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

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