\subsection*{Problem 1}
Let $G$ be a $2$-edge connected graph. Show that the number of ears of even length in any ear decomposition of $G$ has the same parity. In other words, although the ear decomposition is not unique, the parity of the number of even ears is an invariant.

\begin{proof}
\end{proof}

\subsection*{Problem 2}
Assume that all sides of a convex polyhedron are either regular pentagons or regular hexagons. What is the number of pentagons among these sides?

\begin{proof}
\end{proof}

\subsection*{Problem 3}
Prove that the inequality $|E(G)| \geq \binom{\chi(G)}{2}$ holds for every finite simple graph $G$, where $\chi(G)$ denotes the chromatic number of $G$.

\begin{proof}
\end{proof}

\subsection*{Problem 4}
Let $G$ be a $3$-regular graph with edge-chromatic number $3$. Assume that (up to permutation of colors) there is only one valid $3$-edge-coloring of $G$. Prove that $G$ contains a Hamiltonian cycle.

\begin{proof}
\end{proof}

\subsection*{Problem 5}
Prove that $R(4, \ell) \leq \binom{\ell + 2}{3}$ for all $\ell \geq 2$.

\begin{proof}
\end{proof}