Question

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

Answer #1

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

GRAPH THEORY:
Let G be a graph which can be decomposed into Hamilton
cycles.
Prove that G must be k-regular, and that k must be even.
Prove that if G has an even number of vertices, then the edge
chromatic number of G is Δ(G)=k.

Discrete math, graph theory question:
Let G be a graph with 100 vertices, and chromatic number 99.
Prove a lower bound for the clique number of G. (Hint: Any lower
bound will do, but try to make it as large as you can.)

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) 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 e be the unique lightest edge in a graph G. Let T be a
spanning tree of G such that e ∉ T . Prove using elementary
properties of spanning trees (i.e. not the cut property) that T is
not a minimum spanning tree of G.

Let G be a graph in which there is a cycle C odd length that has
vertices on all of the other odd cycles. Prove that the chromatic
number of G is less than or equal to 5.

Exercise 3. Let Wn be the graph obtained from the cycle graph Cn
by adding one new vertex which is adjacent to every vertex of Cn.
Prove that for n ≥ 3, Wn does not have an Eulerian trail.

(a) Show that the chromatic number of the Petersen graph is
exactly 3.
(b) Find the chromatic number of the k-cube Qk for any k, and of
Kn−e where n ≥ 3 and e is any edge

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.

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