MHB Graph Theory Proof: Connectivity of G Proven

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The discussion revolves around proving the connectivity of graph G given that it has exactly two vertices of odd degree and that a new graph H, formed by adding an edge between these two vertices, is connected. The initial proof incorrectly assumed the existence of an Euler walk due to the graph's odd vertices, which only applies to connected graphs. The correct argument establishes that if G is not connected, it must consist of at least two components, each containing an even number of odd vertices, thus forcing the odd vertices x and y to be in the same component. Consequently, adding the edge xy cannot connect disjoint components, implying that if H is connected, G must also be connected. The final conclusion confirms the validity of the proof by contraposition.
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[SOLVED]Graph theory proof

The graph G has precisely two vertices x and y of odd degree. A new graph of multigraph H is formed from G by adding an edge xy. If H is connected, prove that G is connected.

Please check my following proof for this problem.

Since G has exactly two odd vertices, there is an Euler walk between x and y.
Denote this walk by xEy, where E is a string of vertices.

A walk between two vertices in H either (1)contains edge xy or (2)does not contain edge xy.

(1)Let a, b be two vertices connected via edge xy, so aAxyBb, A and B being strings of vertices.
But then there is a walk from a to b in G, by way of aAxEyBb.

(2)If a walk between two vertices that does not contain edge xy exists in H, it also exists in G.

Hence, any two vertices connected in H are connected in G. Therefore, if H is connected, G is connected.
 
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Yuuki said:
The graph G has precisely two vertices x and y of odd degree. A new graph of multigraph H is formed from G by adding an edge xy. If H is connected, prove that G is connected.
Please check my following proof for this problem.
Since G has exactly two odd vertices, there is an Euler walk between x and y.
Denote this walk by xEy, where E is a string of vertices.
A walk between two vertices in H either (1)contains edge xy or (2)does not contain edge xy.
(1)Let a, b be two vertices connected via edge xy, so aAxyBb, A and B being strings of vertices.
But then there is a walk from a to b in G, by way of aAxEyBb.
(2)If a walk between two vertices that does not contain edge xy exists in H, it also exists in G.
Hence, any two vertices connected in H are connected in G. Therefore, if H is connected, G is connected.

Frankly I have no idea how that argument works.

Where have you used "G has precisely two vertices x and y of odd degree"?

If $$G$$ is not connected then it is the union of at least two components.
Because each component is a sub-graph and any graph has an even number of odd verticies, then $$x\&~y$$ belong to the same component.

So how would adding another edge $$xy$$ to $$G$$ connect it?
 
Yuuki said:
The graph G has precisely two vertices x and y of odd degree. A new graph of multigraph H is formed from G by adding an edge xy. If H is connected, prove that G is connected.

Please check my following proof for this problem.

Since G has exactly two odd vertices, there is an Euler walk between x and y.
Denote this walk by xEy, where E is a string of vertices.

A walk between two vertices in H either (1)contains edge xy or (2)does not contain edge xy.

(1)Let a, b be two vertices connected via edge xy, so aAxyBb, A and B being strings of vertices.
But then there is a walk from a to b in G, by way of aAxEyBb.

(2)If a walk between two vertices that does not contain edge xy exists in H, it also exists in G.

Hence, any two vertices connected in H are connected in G. Therefore, if H is connected, G is connected.

I used it here.
But I now realize this is a big mistake, because the argument

a graph has two odd vertices \Rightarrow the graph has an Euler walk

is valid only for connected graphs.
I can't use this assumption because that is precisely what I'm trying to prove.

As for the other argument, I tried to separate it into binary cases.
To use the (wrong) Euler circuit argument.So,

G is not connected.
\Rightarrow There are at least two components.
\Rightarrow Since each component is connected, there must be an even number of odd vertices.
\Rightarrow x and y must belong to the same component because there are only two odd vertices in G.
\Rightarrow Connecting x and y will not connect disjoint components.

Hence, if G is not connected, neither is H, which proves the assumption by contraposition?
 
Yuuki said:
G is not connected.
\Rightarrow There are at least two components.
\Rightarrow Since each component is connected, there must be an even number of odd vertices.
\Rightarrow x and y must belong to the same component because there are only two odd vertices in G.
\Rightarrow Connecting x and y will not connect disjoint components.
Hence, if G is not connected, neither is H, which proves the assumption by contraposition?

That is now correct.
 

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