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{\bf MATH314  \hspace{1cm} HW 6\hfill \name}
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Due \textbf{Mar 3} before class, \textbf{answer without justification will receive 0 points}.
The typing the HW in \LaTeX{} is optional. 
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\question{5.19}{
Prove or disprove: Let $G$ be a nontrivial graph. For every vertex $v$ of $G$, $\kappa(G - v) = \kappa(G)$ or $\kappa(G-v) = \kappa(G) - 1$. (No drawing)
}


\textbf{Solution:}

\question{5.32}{
Show that if $G$ is a connected graph of order at least 3, then its square $G^2$ is 2-connected.
}


\textbf{Solution:}


\question{}{
Let $uv$ be an edge of a graph $G$. \textbf{Subdividing} $uv$ means removing the edge and replacing it with path $uxv$, where $x$ is a new vertex.\\
Show that every 2-connected graph can be obtained from a triangle by adding and subdividing edges.
}


\textbf{Solution:}


\question{}{
Prove that the symmetric difference of two different edge cuts is an edge cut.
(Hint: Draw a picture illustrating the two edge cuts and use it to guide the proof.)
Recall that for sets $A$ and $B$ the symmetric difference of $A$ and $B$ is $(A \setminus B) \cup (B \setminus A)$.
}


\textbf{Solution:}

\question{}{
Let $G$ be a connected $k$-regular graph with $n = 3k$ vertices. Prove that $G$ is (at least) 2-edge-connected.
}


\textbf{Solution:}

\question{}{
Let $G$ be a graph. Let $T$ be a graph whose vertices correspond to blocks in $G$ and two vertices in $T$ are adjacent if the corresponding blocks share a vertex. Show that $T$ is a tree.
}


\textbf{Solution:}




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