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\begin{document}
\begin{center}
{\bf Math 432 Homework Six}\\
Due: Friday March 4, 2016
\end{center}

\bigskip

Prove the following statements.  Four points for each. 

\begin{enumerate}

\item A cycle is a circuit without repeated vertices.  Prove that every $n$-vertex graph with at least $n$ edges contains a cycle. \\


\item Prove or disprove: if $G$ is an Eulerian graph with edges $e,f$ that share a vertex, then $G$ has an Eulerian circuit in which $e,f$ appear consecutively. \\




\item A list of integers $d_1, d_2, \ldots, d_n$ with $d_1\ge d_2\ge \ldots\ge d_n>0$ is graphic if it is the degree list of some simple graph.  Show that \\
(a)   $(7, 6, 5, 4, 3, 3, 2)$ and $(6, 6, 5, 4, 3, 3, 1)$ are not graphic. \\
(b)  if a sequence is graphic, then $\sum_{i=1}^n d_i$ is even and  $\sum_{i=1}^k d_i\le k(k-1)+\sum_{i=k+1}^n \min\{k, d_i\}$ for $1\le k\le n$. \\
(c) (bonus problem) the condition in (b) is also sufficient. \\

\item Show that the Kruskal's algorithm produces a minimum spanning tree in a weighted connected graph.\\

\item We showed in class that every tree is a bipartite graph.  Prove that every tree has a leaf in its larger partite set (in both if they have equal size).  

  

\end{enumerate}
\end{document}
