Question

Let G be a nonempty graph with χ(G) = k . A graph H is obtained from G by subdividing every edge of G. If χ(H) = χ(G), then what is k?

Answer #1

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

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 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

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

Let H, K be two groups and G = H × K. Let H = {(h, e) | h ∈ H},
where e is the identity in K. Show that G/H is isomorphic to K.

Suppose that H is a connected graph that contains a
proper cycle. Let H′ represent the subgraph of H that results by
removing a single edge from H, where the edge removed is part of
the proper cycle that H contains. Argue that H′ remains
connected.
Notes.
Your argument here needs to be (slightly) different from your
argument in Activity 16.3.
Make sure you are using the technical definition of connected
graph in your argument. What are you assuming about...

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!

Let H and K be subgroups of G. Prove that H ∪ K is a subgroup of
G iff H ⊆ K or K ⊆ H.

(Abstract algebra) Let G be a group and let H and K be subgroups
of G so that H is not contained in K and K is not contained in H.
Prove that H ∪ K is not a subgroup of G.

f H and K are subgroups of a group G, let (H,K) be the subgroup
of G generated by the elements {hkh−1k−1∣h∈H, k∈K}.
Show that :
H◃G if and only if (H,G)<H

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