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%% header definitions
\def\course{MATH 314}
\def\assign{HW \#5}
\def\due{Feb 8}
\def\doclink{\url{https://mathematicaster.org/teaching/graphs2022/hw/hw5.tex}}

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\begin{document}
\pagestyle{empty}

\noindent
\begin{minipage}{.33\textwidth}
	\flushleft {\bf \course}
\end{minipage}
\begin{minipage}{.33\textwidth}
	\centering {\bf \assign}
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\begin{minipage}{.33\textwidth}
	\flushright {\bf \due}
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\noindent {\footnotesize This document is from \doclink}
\vskip10pt

\noindent {\small Unless explicitly requested by a problem, do not include sketches as part of your proof.
You are free to use the result from any problem on this (or previous) assignment as a part of your solution to a different problem even if you have not solved the former problem.
}

\vskip20pt


\begin{problem}[0.5 + 0.5 + 1 pts]
    For a fixed integer $k\geq 3$, a graph $G$ is said to have property $\mcal C_k$ if every subgraph of $G$ with at least $k$ edges contains a cycle.
    For a fixed integer $k\geq 4$, a graph $G$ is said to have property $\mcal E_k$ if every subgraph of $G$ with at least $k$ edges contains an even cycle.
    \begin{enumerate}
        \item For each $k\geq 3$, construct a graph $G$ which has property $\mcal C_k$ and has $\abs{E(G)}={k\choose 2}$.
        \item For each $k\geq 4$, construct a graph $G$ which has property $\mcal E_k$ and has $\abs{E(G)}=\lceil k/2\rceil\lfloor k/2\rfloor$.
        \item Prove that if $G$ is a connected graph with property $\mcal C_k$, then $\abs{E(G)}\leq{k\choose 2}$.
    \end{enumerate}
\end{problem}


\begin{problem}[3 pts]
    This problem expands on the observation that trees on at least two vertices have at least two leaves.

    Let $T$ be a tree on at least two vertices.
    Let $\ell(T)$ denote the number of leaves of $T$ and define $D_{\geq 2}=\{v\in V(T):\deg v\geq 2\}$.
    Prove that
    \[
        \ell(T)=2+\sum_{v\in D_{\geq 2}}(\deg v-2)
    \]
\end{problem}



\begin{problem}[3 pts]
    Fix an integer $n\geq 2$.
    Prove that a sequence of integers $d_1,\dots,d_n$ is the degree sequence of some tree if and only if $d_i\geq 1$ for all $i\in[n]$ and $\sum_{i=1}^n d_i=2n-2$.
\end{problem}





\begin{problem}[2 pts]
    Let $G$ be a connected graph and let $w\colon E(G)\to\R$ be a weight function.
    Show that if all weights are distinct (that is $w(e)\neq w(s)$ for all distinct $e,s\in E(G)$), then $G$ has a \emph{unique} minimum spanning tree.
\end{problem}


\end{document}
