path: root/content/graph_theory.tex

\section*{Basics}

A \emph{graph} $G$ is a tuple $(V,E)$ where $V$ is a finite set of vertices and $E \subset {V \choose 2}$ is the set of edges.

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A \emph{multigraph} $G$ is a tuple $(V,E)$ where $V$ is a set of vertices and $E$ is a multiset of elements from ${V \choose 1} \cup {V \choose 2}$ i.e. multiedges and loops.

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A \emph{hypergraph} $H$ is a tuple $(X,E)$ where $X$ is a finite set and $E \subseteq 2^X \setminus \{\emptyset\}$.

i.e. edges may join any number of vertices.

\subsection*{Notation of special graphs}

$K_n$ is the complete graph on $n$ vertices.

$C_n$ is the cycle on $n$ vertices.

$P_n$ is a path of length $n$ where $n$ counts edges.

$E_n$ is the empty i.e. edge-less graph on $n$ vertices.

$K_{m,n}$ is the complete bipartite graph with partite sets of cardinality $m$ respectively $n$.

\subsection*{Local properties}

$N(v)$ for $v \in V(G)$ is the \emph{neighbourhood} of $v$.

$d(v)$ for $v \in V(G)$ is the \emph{degree} of $v$.

$d(u,v)$ is the length of a shortest $u$-$v$-path.

\subsection*{Global properties}

$|G| := |V(G)|$ and $\|G\| := |E(G)|$

$\delta(G)$ is the minimum degree in $G$.

$\Delta(G)$ is the maximum degree in $G$.

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$G$ is $k$-regular $\iff \forall v \in V(G) : d(v) = k$

The \emph{girth} $g(G)$ is the length of a shortest cycle.

The length of a longest cycle is $G$'s \emph{circumference}.

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$\text{diam}(G) := \displaystyle\max_{u,v \in V(G)} d(u,v)$ is the \emph{diameter} of $G$.

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$\text{rad}(G) := \displaystyle\min_{u \in V(G)} \max_{v \in V(G)} d(u,v)$ is the \emph{radius} of $G$.

\subsection*{Handshake Lemma}

$$2|E| = \sum_{v \in V} d(v)$$

Furthermore the sum of all degrees is even and thus \#vertices with odd degree is also even.

\subsection*{Induced Subgraphs}

Subgraph $H \subset G$ is \emph{induced} if:

$\forall u, v \in V(H) : uv \in E(H) \iff uv \in E(G)$

i.e. every induced subgraph may be generated by deleting vertices and their incident edges from $G$.

\subsection*{Bipartite Charaterization}

$G$ is \emph{bipartite} iff it has no cycles of odd length.

\subsection*{Hamiltonian Cycles}

A \emph{Hamiltonian cycle} is a cycle that visits each vertex of $G$ exactly once.

Correspondingly a \emph{Hamiltonian path} is a path that visits each vertex of $G$ exactly once.

\subsubsection*{Dirac Theorem}

Every graph $G$ with $|V(G)| \geq 3$ and $\delta(G) \geq \frac{n}{2}$ has a Hamiltonian cycle.

\subsection*{Eulerian Tour Condition}

An \emph{Eulerian tour} is closed walk containing all edges of $G$ exactly once.

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A connected graph has an Eulerian tour iff every vertex has even degree.

\subsection*{Hall's Marriage Theorem}

Bipartite $G$ with partite sets $A, B$ has a matching containing $A$ iff $\forall S \subset A : |N(S)|\geq|S|$.

\subsection*{Tutte's Theorem}

Let $q(G)$ be the number of odd components in $G$.

$G$ has perfect matching iff $\forall S \subseteq V : q(G-S) \leq |S|$.

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Any $k$-regular bipartite graph has a perfect matching and a proper $k$-edge coloring.

\subsection*{König's Theorem}

For bipartite $G$ the size of the smallest matching equals the size of a smallest vertex cover.

\subsection*{Hajnal and Szemer\'{e}di}

Let $H, G$ be graphs. An \emph{$H$-factor} of $G$ is a spanning subgraph that is a set of disjoint copies of $H$ in $G$ whose vertex sets form a partition of $V(G)$.

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$\delta(G) \geq (1-\frac{1}{k})n$ for $k | n \implies G$ has a $K_k$-factor.

\section*{Connectivity}

Let $\kappa(G)$ be the \emph{connectivity} of $G$

i.e. the maximum $k$ for which $G$ is \emph{$k$-connected}.

$G$ is $k$-connected if $k-1$ vertices can be removed without disconnecting.

Let $\kappa'(G)$ be the edge-connectivity of $G$:

\vspace*{-3mm}
$$\kappa(G) \leq \kappa'(G) \leq \delta(G)$$

\subsection*{Menger's Theorem}

For $A,B \subseteq V(G)$ the min \#vertices separating $A, B$ equals the max number of disjoint $A$-$B$-paths.

For $a, b \in V(G)$ s.t. $\{a,b\} \notin E(G)$ the minimum number of $v \in V(G)\setminus\{a,b\}$ separating $a, b$ equals the maximum number of independent $a$-$b$-paths.

\subsubsection*{Global Menger's Theorem}

Graph $G$ is $k$-connected iff $\forall a,b \in V(G)$ there exist $k$ independent $a$-$b$-paths.

\subsection*{Ear-decomposition}

Ear-decomposition of $G$ is a sequence of graphs $G_0 \subseteq G_1 \subseteq \cdots \subseteq G_k$ s.t. $G_0$ is a cycle, $G_i$ results from $G_{i-1}$ by adding an ear and $G_k = G$.

$G$ is $2$-connected iff it has an ear-decomposition.

\subsection*{Block-cut-vertex-graph}

Blocks of $G$ are the maximal $2$-connected subgraphs and bridges.

The \emph{block-cut-vertex-graph} of $G$ is the bipartite graph s.t. its partite sets are the blocks on the one side and the cut-vertices on the other side.

Block-cut-vertex-graph of connected $G$ is a tree.

\section*{Planar Graphs}

Let a \emph{plane graph} be a set of points on the plane connected by arcs s.t. the arcs do not contain any of the points or intersect any other arc.

A \emph{planar embedding} is a bijection between a plane graph and a abstract graph.
A \emph{planar graph} is a graph $G$ with a planar embedding. The corresponding plane graph is a drawing of $G$.

\subsection*{F\'{a}ry's Theorem}

Every planar graph can be embedded in a plane s.t. the edges are straight lines.

\subsection*{Euler's Formula}

For connected planar $G$ with $v$ vertices, $e$ edges and $f$ faces: $v-e+f=2$

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$|E(G)| \leq 3|V(G)|-6$ (equal if $G$ is triangulation)

$|E(G)| \leq 2|V(G)|-4$ if no face is bound by triangle.

\subsection*{Tutte's Theorem}

Every $4$-connected planar graph is Hamiltonian.

\subsection*{Graph minors}

$H$ is \emph{minor} of $G$ (i.e. $MH \subseteq G$) if it can be generated from $G$ by deleting vertices, deleting or contracting edges.

$H$ is a \emph{subdivision} of $G$ if it can be generated from $G$ by subdividing edges.

$H$ is a \emph{topological minor} of $G$ (i.e. $TH \subseteq G$) if a subgraph of $G$ is a subdivision of $H$.

\subsubsection*{Kuratowski's Theorem}

For graph $G$ the following is equivalent:

\begin{enumerate}[label=(\alph*)]
	\item $G$ is planar
	\item $K_5, K_{3,3}$ aren't minors of $G$
	\item $K_5$, $K_{3,3}$ aren't topological minors of $G$
\end{enumerate}


\section*{Colorings}

The \emph{chromatic number} $\chi(G)$ and the \emph{chromatic index} $\chi'(G)$ are defined as follows:

\vspace*{-4mm}
\begin{align*}
\chi(G) :&= \min_{k \in \N} : G \text{ has proper $k$-vertex coloring} \\
\chi'(G) :&= \min_{k \in \N} : G \text{ has proper $k$-edge coloring}
\end{align*}

$k$-regular bipartite $G$ has proper $k$-edge-coloring.

\subsection*{Cliques}

A \emph{clique} is a subgraph of $G$ that is a complete graph.
The \emph{clique number} $\omega(G)$ is the maximum order of a clique in $G$.
The \emph{co-clique number} $\alpha(G)$ is the largest order of an independent set in $G$.

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$G$ is \emph{perfect} if $\forall$ induced $H \subseteq G : \chi(H) = \omega(H)$.

Bipartite graphs are perfect with $\chi = \omega = 2$.

\subsection*{$5$-Color Theorem}

Every planar graph is $5$-colorable.

\subsection*{List coloring}

$\forall v \in V(G)$ let $L(v) \subseteq \N$ be a list of colors.

$G$ is \emph{$L$-list-colorable} if $\exists$ a proper coloring $c$ s.t. $\forall v \in V(G) : c(v) \in L(v)$.

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$G$ is \emph{$k$-list-colorable} or \emph{$k$-choosable} if it is $L$-list-colorable for all lists of $k$ colors.

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\emph{Choosability} $ch(G) := \min_{k\in\N}\{ G$ is $k$-choosable$\}$

\subsubsection*{Thomassen's $5$-List-Color Theorem}

Every planar graph is $5$-choosable.

\subsection*{Greedy chromatic number estimate}

$$\chi(G) \leq \Delta(G)+1$$

\subsection*{Brook's Theorem}

If $G$ is connected and neither complete nor an odd cycle then $\chi(G) \leq \Delta(G)$

\subsection*{König's Theorem}

$$G \text{ bipartite} \implies \chi'(G) = \Delta(G)$$

\subsection*{Vizing's Theorem}

$$\chi'(G) \in \{\Delta(G), \Delta(G)+1\}$$

\section*{Extremal Graph Theory}

For $n \in \N$ and graph $H$ the \emph{extremal number} $ex(n,H)$ is the max number of edges in a gra