From f84c55b21cac57f69cb794ee4c95259405903a45 Mon Sep 17 00:00:00 2001 From: Adrian Kummerlaender Date: Tue, 13 Feb 2018 16:59:29 +0100 Subject: Reorder, expand graph theory digest --- content/graph_theory.tex | 89 +++++++++++++++++++++++++++--------------------- 1 file changed, 51 insertions(+), 38 deletions(-) (limited to 'content') diff --git a/content/graph_theory.tex b/content/graph_theory.tex index 50b28dd..8975109 100644 --- a/content/graph_theory.tex +++ b/content/graph_theory.tex @@ -12,7 +12,6 @@ A \emph{hypergraph} $H$ is a tuple $(X,E)$ where $X$ is a finite set and $E \sub i.e. edges may join any number of vertices. - \subsection*{Notation of special graphs} $K_n$ is the complete graph on $n$ vertices. @@ -57,6 +56,12 @@ $\text{diam}(G) := \displaystyle\max_{u,v \in V(G)} d(u,v)$ is the \emph{diamete $\text{rad}(G) := \displaystyle\min_{u \in V(G)} \max_{v \in V(G)} d(u,v)$ 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: @@ -65,15 +70,19 @@ $\forall u, v \in V(H) : uv \in E(H) \iff uv \in E(G)$ i.e. every induced subgraph may be generated by deleting vertices and their incident edges from $G$. -\subsection*{Handshake Lemma} +\subsection*{Bipartite Charaterization} -$$2|E| = \sum_{v \in V} d(v)$$ +$G$ is \emph{bipartite} iff it has no cycles of odd length. -Furthermore the sum of all degrees is even and thus \#vertices with odd degree is also even. +\subsection*{Hamiltonian Cycles} -\subsection*{Bipartite Charaterization} +A \emph{Hamiltonian cycle} is a cycle that visits each vertex of $G$ exactly once. -$G$ is \emph{bipartite} iff it has no cycles of odd length. +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} @@ -111,21 +120,20 @@ $\delta(G) \geq (1-\frac{1}{k})n$ for $k | n \implies G$ has a $K_k$-factor. \section*{Connectivity} -Let $\kappa(G)$ be the connectivity of $G$ i.e. the maximum $k$ for which $G$ is $k$-connected. +Let $\kappa(G)$ be the \emph{connectivity} of $G$ -$G$ is $k$-connected if $k-1$ vertices can be removed without disconnecting. +i.e. the maximum $k$ for which $G$ is \emph{$k$-connected}. -Let $\kappa'(G)$ be the edge-connectivity of $G$. +$G$ is $k$-connected if $k-1$ vertices can be removed without disconnecting. -For any non-trivial connected $G$: +Let $\kappa'(G)$ be the edge-connectivity of $G$: +\vspace*{-3mm} $$\kappa(G) \leq \kappa'(G) \leq \delta(G)$$ \subsection*{Menger's Theorem} -For graph $G$ and vertex sets $A,B \subseteq V(G)$ the minimum number of vertices separting $A$ and $B$ equals the maximum number of disjoint $A$-$B$-paths. - -\spacing +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. @@ -137,8 +145,6 @@ Graph $G$ is $k$-connected iff $\forall a,b \in V(G)$ there exist $k$ independen 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 $G_k = G$. -\spacing - $G$ is $2$-connected iff it has an ear-decomposition. \subsection*{Block-cut-vertex-graph} @@ -147,8 +153,6 @@ 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. -\spacing - Block-cut-vertex-graph of connected $G$ is a tree. \section*{Planar Graphs} @@ -172,16 +176,16 @@ $|E(G)| \leq 3|V(G)|-6$ (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. -\spacing - $H$ is a \emph{subdivision} of $G$ if it can be generated from $G$ by subdividing edges. -\spacing - $H$ 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} @@ -197,6 +201,8 @@ For graph $G$ the following is equivalent: \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} \\ @@ -209,6 +215,7 @@ $k$-regular bipartite $G$ has proper $k$-edge-coloring. 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 $G$. \spacing @@ -230,7 +237,7 @@ $G$ is \emph{$L$-list-colorable} if $\exists$ a proper coloring $c$ s.t. $\foral $G$ is \emph{$k$-list-colorable} or \emph{$k$-choosable} if it is $L$-list-colorable for all lists of $k$ colors. -\subsubsection*{Choosability} +\spacing \emph{Choosability} $ch(G) := \min_{k\in\N}\{ G$ is $k$-choosable$\}$ @@ -248,7 +255,7 @@ If $G$ is connected and neither complete nor an odd cycle then $\chi(G) \leq \De \subsection*{König's Theorem} -$G$ bipartite $\implies \chi'(G) = \Delta(G)$ +$$G \text{ bipartite} \implies \chi'(G) = \Delta(G)$$ \subsection*{Vizing's Theorem} @@ -373,6 +380,7 @@ e.g. $R(2,k) = R(k,2) = k$ and $R(3) = 6$ \subsection*{Ramsey Theorem} +\vspace*{-2mm} $$\forall k \in \N : \sqrt{2}^k \leq R(k) \leq 4^k$$ Particularly Ramsey, asymmetric Ramsey and graph Ramsey numbers are finite. @@ -395,14 +403,14 @@ $\implies \exists x, y, z \in \N$ of the same color s.t. $x+y=z$. Let $G$ be a multigraph, $T := \{(x,e,y) | xy = e \in E(G)\}$. -$f : T \to \mathbb{K}$ defines a \emph{circulation} on $G$ if: +For abelian group $H$ the function $f : T \to H$ defines a \emph{circulation} on $G$ if: \begin{enumerate} \item $\forall (x,e,y) \in T, x \neq y : f(x,e,y) = -f(y,e,x)$ \item $\forall v \in V(G) : f(v,V) = 0$ \end{enumerate} -$f(X,Y) := \displaystyle\sum_{(x,e,y) \in T, x \neq y, x \in X, y \in Y} f(x,y)$ +Let $f(X,Y) := \sum_{(x,e,y) \in T, x \neq y, x \in X, y \in Y} f(x,y)$ \spacing @@ -440,7 +448,7 @@ $(S,\overline S)$ for $S \subseteq V(G)$ s.t. $s \in S, t \notin S, \overline S \spacing -Its \emph{capacity} is $c(S,\overline S) := \displaystyle\sum_{x \in S, y \in \overline S, (x,e,y) \in T} c(x,y)$. +Its \emph{capacity} is $c(S,\overline S) := \sum_{x \in S, y \in \overline S, (x,e,y) \in T} c(x,y)$. \spacing @@ -448,9 +456,9 @@ For any flow $f$ and cut $(S,\overline S) : f(S,\overline S) = f(s,V(G))$. \subsubsection*{Ford-Fulkerson Theorem} -In any network the max value of a flow equals the min capacity of a cut. +In any network the maximum value of a flow equals the minimum capacity of a cut. -There is an integral flow with this max flow value. +An integral flow $f$ with this max flow value exists. \section*{Random Graphs} @@ -464,6 +472,7 @@ A \emph{property} $\mathcal{P}$ is a set of graphs. $G \in \mathcal{G}(n,p_n)$ \emph{almost always} has property $\mathcal{P}$ if: +\vspace*{-2mm} $$\mathbb{P}(G \in \mathcal{G}(n,p_n) \cap \mathcal{P}) \to 1 (n \to \infty)$$ If $p_n$ is constant it is said that \emph{almost all} $G \in \mathcal{G}(n,p)$ have property $\mathcal{P}$. @@ -479,6 +488,13 @@ $f : \N \to [0,1]$ is called \emph{threshold function} for $\mathcal{P}$ if: Such functions do not exist for all properties $\mathcal{P}$. +\subsection*{Simple probabilities} + +Let $G$ be a graph on $n$ vertices and $m$ edges. + +\vspace*{-2mm} +$$\mathbb{P}(G = \mathcal{G}(n,p)) = p^m (1-p)^{{n \choose 2}-m}$$ + \subsection*{Expected values and indicators} Let $X : \mathcal{G}(n,p) \to \N$ be a random variable. @@ -488,20 +504,17 @@ When $X$ counts e.g. the number of a certain class of subgraphs $\mathbb{E}(X)$ \vspace*{-4mm} $$\mathbb{E}X = \mathbb{E}\left(\sum_{i=1}^N X_i\right) = \sum_{i=1}^N \mathbb{E}X_i = \sum_{i=1}^N \mathbb{P}(X_i=1)$$ -\subsection*{Erd\H{o}s-Hajnal} +\subsubsection*{Expected number of $k$-cycles} -$\forall k \geq 3 \exists$ a graph $G$ s.t. $g(G), \chi(G) > k$. +$$\mathbb{E}(\#k\text{-cycles in } G \in \mathcal{G}(n,p)) = \frac{n_k}{2k} \cdot p^k$$ -\section*{Hamiltonian Cycles} +for $n_k := n \cdot (n-1) \cdots (n-k+1)$. -A \emph{Hamiltonian cycle} is a cycle that visits each vertex of $G$ exactly once. +\subsection*{Erd\H{o}s' lower bound on $R(k,k)$} -Correspondingly a \emph{Hamiltonian path} is a path that visits each vertex of $G$ exactly once. - -\subsection*{Dirac Theorem} +$$R(k.k) \geq 2^{k/2}$$ -Every graph $G$ with $|V(G)| \geq 3$ and $\delta(G) \geq \frac{n}{2}$ has a Hamiltonian cycle. +\subsection*{Erd\H{o}s-Hajnal} -\subsection*{Tutte Theorem} +$\forall k \geq 3 \exists$ a graph $G$ s.t. $g(G), \chi(G) > k$. -Every $4$-connected planar graph is Hamiltonian. -- cgit v1.2.3