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how to test if a graph is a tree

I have the following question:

Let $X$ be a finite set and let $G=(V,E)$ be a graph such that $V =
> S \cup T$ and $E = E_1 \cup E_2$. Let $G^* = (S, E_1 \cup
> E_2)$, and $G^0 = (S \setminus T, E_1 \cup E_2)$.
1.
A) Show that $\chi(G^*) \leq |S|$.
2.
B) For any graph $G=(V,E)$, $G$ is a tree if and only if $G$ is
regular and $\chi(G^*) = |V| – 1$.

I have trouble with the second part, to be more precise with 1).
My ideas:
«$\Rightarrow$»:
Since $G^*$ is a supergraph of $G$ then it is also a spanning subgraph of $G$ that means that it is a spanning subgraph of $G$. And since $G^*$ is also regular then it follows that $G$ is regular.
«$\Leftarrow$»:
Assume that $G$ is regular, then its vertices can be partitioned such that $V = V_1 \cup V_2$ with $V_1 \cap V_2 = \emptyset$ such that all vertices have the same degree. Now let $S’ = \{v_1\}$ and $T’ = \emptyset$. Then $G^* = (S’, E_1)$ and the sets $V_1$ and $V_2$ are subsets of $S$ and $T$ respectively. Since all vertices have the same degree in $G$, then the subsets $V_1$ and $V_2$ are also independent and both have the same size. However, $S
eq V_1 \cup V_2$

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MS Riverbird is a replica of the original paddle-steamer City of New Orleans (1885), which travelled the Mississippi River between St Louis and New Orleans. The original City of New Orleans had been built in 1854 by the New Orleans Steamboat and Transportation Company, which later became a subsidiary of the American Home Life Insurance Company. The replica was built between 1997 and 1998 by Macgregor Marine of Georgia, under contract to the American Home Life Insurance Company.

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See also

Great Circle Route

References

Category:MV Riverbird
Category:Culture of New Orleans
Category:Riverboats in Louisiana
Category:Mississippi River
Category:New Orleans
Category:Replica ships
Category:Ships built in LouisianaThe English Teacher (film)

The English Teacher is a 1958 East German drama film directed by Helmut Weiss and starring Ingrid Dorn, Heinz Müller and Erna Sellmer. It was one of the first social realist films from East Germany.

Cast
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Fritz Nagel as Prewiegel
Paul Dahlke as Erster Redakteur
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References

Bibliography