Add A&D summary. Still WIP for fixing errors

algorithms-and-datastructures/parts/graphs/paths-walks-cycles.tex

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+\newpage
+\subsection{Paths, Walks, Cycles}
+\begin{definition}[]{Paths, Walks, and Cycles in Graphs}
+    \begin{itemize}
+        \item \textbf{Walk:} A sequence of vertices and edges in a graph, where each edge connects consecutive vertices in the sequence. Vertices and edges may repeat.
+        \item \textbf{Path:} A walk with no repeated vertices. In a directed graph, the edges must respect the direction.
+        \item \textbf{Cycle:} A path that starts and ends at the same vertex. For a simple cycle, all vertices (except the start/end vertex) and edges are distinct.
+        \item \textbf{Eulerian Walk:} A walk that traverses every edge of a graph exactly once.
+        \item \textbf{Closed Eulerian Walk (Eulerian Cycle):} An Eulerian walk that starts and ends at the same vertex.
+        \item \textbf{Hamiltonian Path:} A path that visits each vertex of a graph exactly once. Edges may or may not repeat.
+        \item \textbf{Hamiltonian Cycle:} A Hamiltonian path that starts and ends at the same vertex.
+    \end{itemize}
+\end{definition}
+
+\begin{properties}[]{Eulerian Graphs}
+    \begin{itemize}
+        \item \textbf{Undirected Graph:}
+              \begin{itemize}
+                  \item A graph has an Eulerian walk if it has exactly two vertices of odd degree (necessary condition).
+                  \item A graph has a closed Eulerian walk if all vertices have even degree (necessary condition).
+              \end{itemize}
+        \item \textbf{Directed Graph:}
+              \begin{itemize}
+                  \item A graph has an Eulerian walk if at most one vertex has \textit{in-degree} one greater than its \textit{out-degree}, and at most one vertex has \textit{out-degree} one greater than its \textit{in-degree}. All other vertices must have equal in-degree and out-degree.
+                  \item A graph has a closed Eulerian walk if every vertex has equal in-degree and out-degree.
+              \end{itemize}
+    \end{itemize}
+\end{properties}
+
+\begin{properties}[]{Hamiltonian Graphs}
+    \begin{itemize}
+        \item Unlike Eulerian walks, there is no simple necessary or sufficient condition for the existence of Hamiltonian paths or cycles.
+        \item A graph with $n$ vertices is Hamiltonian if every vertex has a degree of at least $\lceil n / 2 \rceil$ (Dirac's Theorem, sufficient condition).
+    \end{itemize}
+\end{properties}
+
+\begin{example}[]{Eulerian and Hamiltonian}
+    \begin{center}
+        \begin{tikzpicture}[node distance=1.5cm, main/.style={circle, draw, fill=blue!20, minimum size=10mm, inner sep=0pt}] % Eulerian graph example
+            \node[main] (1) {1};
+            \node[main] (2) [above right of=1] {2};
+            \node[main] (3) [below right of=1] {3};
+            \node[main] (4) [right of=3] {4};
+            \node[main] (5) [above right of=4] {5};
+            \draw (1) -- (2) -- (5) -- (4) -- (3) -- (1) -- (4) -- (2);
+            \draw (2) -- (3);
+
+            % Hamiltonian graph example
+            \node[main] (6) [right of=5, xshift=3cm] {A};
+            \node[main] (7) [above right of=6] {B};
+            \node[main] (8) [below right of=6] {C};
+            \node[main] (9) [right of=7] {D};
+            \node[main] (10) [right of=8] {E};
+            \draw (6) -- (7) -- (9) -- (10) -- (8) -- (6);
+            \draw (6) -- (10);
+            \draw (7) -- (8);
+        \end{tikzpicture}
+    \end{center}
+\end{example}
+
+\begin{remarks}[]{Key Differences Between Eulerian and Hamiltonian Concepts}
+    \begin{itemize}
+        \item Eulerian paths are concerned with traversing every \textbf{edge} exactly once, while Hamiltonian paths are about visiting every \textbf{vertex} exactly once.
+        \item Eulerian properties depend on the degree of vertices, whereas Hamiltonian properties depend on overall vertex connectivity.
+    \end{itemize}
+\end{remarks}