01ZTGA:Kapitola1 6

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Součásti dokumentu 01ZTGA

součást	akce		poslední editace			soubor
Hlavní dokument	editovat	Hlavní stránka dokumentu 01ZTGA	Karel.brinda	15. 1. 2012	23:45
Řídící stránka	editovat	Definiční stránka dokumentu a vložených obrázků	Admin	7. 9. 2015	13:51
Header	editovat	Hlavičkový soubor	Karel.brinda	15. 1. 2012	12:34	header.tex
Kapitola0	editovat	Úvod	Karel.brinda	15. 1. 2012	12:36	cast0.tex
Kapitola1_1	editovat	Základní pojmy	Karel.brinda	15. 1. 2012	12:46	cast1_kapitola1.tex
Kapitola1_2	editovat	Souvislost	Karel.brinda	15. 1. 2012	12:49	cast1_kapitola2.tex
Kapitola1_3	editovat	Bipartitní grafy	Karel.brinda	15. 1. 2012	12:50	cast1_kapitola3.tex
Kapitola1_4	editovat	Stromy	Kubuondr	5. 1. 2019	09:06	cast1_kapitola4.tex
Kapitola1_5	editovat	Hledání minimální kostry grafu	Karel.brinda	15. 1. 2012	12:51	cast1_kapitola5.tex
Kapitola1_6	editovat	Jednotažky	Karel.brinda	15. 1. 2012	12:53	cast1_kapitola6.tex
Kapitola1_7	editovat	Hamiltonovské kružnice a grafy	Karel.brinda	15. 1. 2012	13:34	cast1_kapitola7.tex
Kapitola1_8	editovat	Párování v grafech	Karel.brinda	15. 1. 2012	13:40	cast1_kapitola8.tex
Kapitola1_9	editovat	Toky v sítích	Karel.brinda	15. 1. 2012	13:44	cast1_kapitola9.tex
Kapitola1_10	editovat	Hranové obarvení grafu	Karel.brinda	15. 1. 2012	13:48	cast1_kapitola10.tex
Kapitola1_11	editovat	Vrcholové obarvení grafu	Karel.brinda	15. 1. 2012	13:52	cast1_kapitola11.tex
Kapitola1_12	editovat	Planární grafy	Karel.brinda	15. 1. 2012	13:56	cast1_kapitola12.tex
Kapitola1_13	editovat	Vlastní čísla adjacenční matice grafu	Karel.brinda	15. 1. 2012	13:57	cast1_kapitola13.tex
Kapitola2_1	editovat	Brouwerova věta o pevném bodě	Karel.brinda	15. 1. 2012	14:11	cast2_kapitola1.tex
Kapitola2_2	editovat	Pravděpodobnostní důkazy v teorii grafů	Karel.brinda	15. 1. 2012	14:12	cast2_kapitola2.tex
Kapitola2_3	editovat	Extremální teorie grafů	Karel.brinda	15. 1. 2012	14:16	cast2_kapitola3.tex
Kapitola2_4	editovat	Ramseyovská čísla	Karel.brinda	15. 1. 2012	14:18	cast2_kapitola4.tex
Kapitola3_1	editovat	Obyčejné mocninné řady	Karel.brinda	15. 1. 2012	14:22	cast3_kapitola1.tex
Kapitola3_2	editovat	Exponenciální generující funkce	Karel.brinda	15. 1. 2012	14:22	cast3_kapitola2.tex
Zdrojový kód

%\wikiskriptum{01ZTGA}
 
\section{Jednotažky}
 
Název kapitoly neformálně vystihuje vlastnost tzv. eulerovských grafů,
které je možné nakreslit jedním tahem. Vše, co bude řečeno o eulerovských
grafech, lze aplikovat i na zobecněné grafy ve smyslu definice \ref{def:zobecneny_graf}.
 
\begin{notation}
Buď $G=(V,E)$ graf, $v_{0},v_{1},...,v_{k}$ sled v $G$. Potom tento
sled zapisujeme také jako $v_{0}e_{1}v_{1}e_{2}v_{2}...e_{k}v_{k}$,
přičemž $\left(\forall i\in\{1,2,...,k\}\right)\left(e_{i}=\{ v_{i-1},v_{i}\}\right).$
\end{notation}
\begin{defn}
Graf $G=(V,E)$, resp. $G=(V,E,\varphi)$, se nazývá \textbf{eulerovský}
(angl. \emph{eulerian}), existuje-li v něm ,,eulerovský{}`` cyklus
(angl. \emph{Euler tour}) $v_{0}e_{1}v_{1}e_{2}v_{2}...e_{m}v_{m}$
takový, že \[
\left(\forall i,j\in\{1,2,...,m\}\right)\left(i\neq j\Rightarrow e_{i}\neq e_{j}\}\right)\]
 a $E=\{ e_{1},...,e_{m}\}$, tj. $m=\# E$.
\end{defn}
\begin{rem*}
Řekneme, že sled $v_{0}e_{1}v_{1}e_{2}v_{2}...e_{k}v_{k}$ je \textbf{tah}
(angl. \emph{trail}) v $G$, jestliže \[
\left(\forall i,j\in\{1,2,...,k\}\right)\left(i\neq j\Rightarrow e_{i}\neq e_{j}\}\right).\]
 
\end{rem*}
\begin{thm}
\label{thm:eulerovske-grafy}Buď $G=(V,E)$ souvislý graf. Potom $G$
je eulerovský, právě když $\left(\forall v\in V\right)(d(v)$ je sudý$)$.
\end{thm}
\begin{proof}
$\boxed{{\Rightarrow:}}$
 
$G$ je eulerovský, v $G$ tedy existuje cyklus, který projde všechny
hrany, a to každou právě jednou. Půjdeme-li po tomto cyklu, je zřejmé,
že vstoupíme do každého vrcholu právě tolikrát, kolikrát z něj vystoupíme,
a to nikdy po hraně, po které jsme již prošli. Z toho plyne, že na
každý vrchol je napojen sudý počet hran.
 
$\boxed{{\Leftarrow:}}$
 
Když má každý vrchol sudý stupeň, tak jeden ($v_{1}$) vybereme a
vydáme se po libovolné hraně, která z něj vede. Z vrcholu, do nějž
jsme se dostali, pokračujeme stejným způsobem dál. Přitom za sebou
obarvujeme hrany a nikdy se nevydáme po hraně, která je již obarvená.
Je zřejmé, že jediný vrchol, z nějž už nebudeme schopni pokračovat
dál, je ten, ze kterého jsme začínali. Potom už jsme buď prošli všechny
hrany, nebo z několika vrcholů vede nenulový, ale sudý počet dosud
neobarvených hran. Vybereme jeden ($v_{2}$) takový, který leží na
cyklu, který jsme již obarvili (to musí být možné, jinak by graf nebyl
souvislý). Z něj začneme nový cyklus. Po jeho dokončení oba cykly
sjednotíme, a to tak, že původní cyklus začneme ve $v_{1}$, přerušíme
jej ve $v_{2}$, provedeme druhý cyklus, a následně dokončíme cyklus
původní. Úvahu lze opakovat, dokud existují neobarvené hrany. Právě
popsaný postup je znázorněn na obrázku \ref{cap:eulerovska_cesta}.%
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\caption{\label{cap:eulerovska_cesta}Tvorba cyklu v eulerovském grafu}
\end{figure}
 
\end{proof}
\begin{rem*}
Existují také tzv. náhodně eulerovské grafy, které mají jeden vrchol
s tou vlastností, že při náhodném průchodu grafu a barvení cest za
sebou lze vždy pokračovat po neobarvených hranách až na případ, kdy
se nacházíme ve startovním vrcholu a všechny hrany už jsou obarvené.
\end{rem*}
 
 
\begin{rem*}
Uvažujme jednotažky takové, že je možné je namalovat jedním tahem
a přitom začít a skončit v obecně různých vrcholech. Tyto jednotažky
jsou právě takové souvislé grafy, které splňují jednu z následujících
dvou podmínek (viz obrázek \ref{cap:jednotazka}):%
\begin{figure}
\begin{center}
%Title: jednotazka.fig
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\end{center}
 
 
\caption{\label{cap:jednotazka}Jednotažka se startovním a cílovým vrcholem}
\end{figure}
 
\begin{enumerate}
\item Všechny vrcholy mají sudý stupeň.
\item Právě dva vrcholy mají lichý stupeň.
\end{enumerate}
\end{rem*}