To prove that, a graph $G$ is bi-colored iff, when in it doesn't have a cycle of odd lengths. To prove that, a graph $G$ is bi-colored iff, when in it doesn't have a cycle of odd lengths. How to prove this theorem, I need some hints/resource recommendation.
Graph with Cycle and Two-Colorable i think if the graph G has an odd cycle, it's not two-colorable, otherwise it can be two colorable.   i read in one notes that the following is True: we couldent two-colorable any graph G that has cycle.    anyone could clarify me ?
I'm relatively new to ring theory so this is probably a simple question.  Its easy to see that a finite ring that is commutative and has no-zero divisors (i.e. an integral domain) must have multiplicative inverses.  I am wondering if we can rearrange these properties and always get implication or produce finite rings with only 2 of the above properties.  Explicitly my questions are:  1) Does a finite ring that is commutative with multiplicative inverses always have no-zero divisors? (EDIT: this one is pretty easy too, let $xy=0$ and assume $x\neq 0$. then $x^{-1}xy=x^{-1}0$. Hence, $y=0$ as desired.)  2) Is a finite ring that has multiplicative inverses and no zero divisors always commutative?  If these are actually simple exercises, I'd just like a hint to get started with the proof or any counterexamples.  Thanks :) How do you make samba follow symlink outside the shared path This is Ubuntu server 10.04 64 and samba 3.4.7.  I have a shared directory /home/mit/share and another one /home/temp that I link into the shared one:  ln -s /home/temp /home/mit/share/temp   But on windows, after using internet, I cannot open S:/temp, but on Linux it is possible to access /home/mit/share/temp like expected.   This works if I link directories inside /home/mit/share/temp, so I guess samba is restricting to jump with a link outside/above the shared directory.   EDIT:  See also this question titled .  It seems best to put unix extensions = no into the global section and follow symlinks = yes and wide links = yes only into the shares section, where you really need it.  The unix extension flag has to live in the global section and not in the individual shares sections. But for security reasons it is better to use the other options only where you need it, and not globally. Creating compressed GeoTIFF with QGIS Clipper tool? I am unable to print documents using my HP printer. However, I am able to print test pages.  What I tried:  I have tried removing and re-installing the printer from the system settings but I can only print once (the first task). Then all the following tasks get stuck at the "Rendering Completed" status.  I additionally tried restarting the computer which leads in the same task mentioned above.  Moreover, I logged into the CUPS system ( 1.6.2) from http://localhost:631  From there, I removed the printer, restarted it and re-installed with no sucess.  System Specifications:  System : Ubuntu GNOME 13.04  Memory: 3.9 GB  CPU : AMD Athlon(tm) 64 X2 Dual Core Processor 4200+ × 2   OS-type : 32 bit  Graphics: Gallium 0.4 on NVA3  Printer: HP Deskjet 2050 j510 Series  IP Adress: localhost  I had no problems with my previous OS which was Pinguy OS 11.04 ( a distro based on Ubuntu 11.04) and it's the first time I use my printer at this OS.  Screenshots:   My printer stuck at "Rendering Completed" status.     From the CUPS web interface; you can see that "Rendering Completed" is stuck at the state of the print process.  **EDIT:**I tried printing from the pdf.js pdf viewer of firefox: I wait for about 5 minutes for a 10 pages document to print, the document is printed like faded and sort of in bold font (I accuse rendering for that). Then the status of the printer gets stuck at 'Rendering Completed' but I can print later on. Again, for the next prints I wait for 5 minutes minimum ( I wait for the 'Rendering....' bar to fill ) and the printing is weird. What animal is a “weefil”? What animal is depicted in this image labelled “weefil”? What is wrong with defining \bal as \begin{align}? The strange thing about the maximum in Planck's law Correct use of will & would? What would be the correct use of will &amp; would in these sentences?  1) What will happen if I say to my boss that I will not come tomorrow?  2) What will happen if I would say to my boss that I will not come tomorrow?  3) What will happen if I will say to my boss that I will not come tomorrow?  4) What would happen if I would say to my boss that I will not come tomorrow?  There are four sentences which one should I use and in which situation? I was looking at  and I didn't realize that Minecraft had some Easter eggs in it. Besides the Christmas chest Easter egg, and the sheep Easter egg, what else is there? Prove that $p \in \mathbb{R}[x]$ can be represented as a sum of squares of polynomials from $\mathbb{R}[x]$ How do I get a book graphic and description from the Amazon Book API? sends an ISBN number to Amazon and gets back a small bit of XML including author, title, and publisher.  However, I also want to get small, medium and large graphic and book descriptions of the title.  Problem: I can find no REST URL examples/documention that work, either at Google or when logged into my "AWS Account" at Amazon Associates.  I find a lot of examples from 2003-2005 but they are all out-of-date and give errors, it seems that Amazon's cloud web services have obfuscated their simple REST API documentation for their books.  Can anyone point me to some documentation on how I can get detailed information about books at Amazon via REST/XML?   Here's what I have tried so . How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$? I'd like to find out why  \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6 \end{align}  I tried to rewrite it into a geometric series  \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = \sum_{n=0}^{\infty} \Big(\frac{1}{2}\Big)^nn^2 \end{align}  But I don't know what to do with the $n^2$. Parallel.ForEach Slower than ForEach $H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$ Given the following definition of normal subgroup     A subgroup $H$ of a group $G$ is said to be normal if, for every $g\in G$: $$gH=Hg$$   I've tried to show that $H\mathrel{\unlhd} G$ if and only if we have $gHg^{-1}=H$ for every $g\in G$:  (if) Let $H\mathrel{\unlhd} G$, $g\in G$; if $l\in gHg^{-1}$, there exists $h\in H$ such that $l=ghg^{-1}$ and since $gh$ = $h'g$ for some $h'\in H$, then $l=ghg^{-1}=h'(gg^{-1}) = h'\in H$. Let $h\in H$; we can write $h$ as $hgg^{-1}$, and since $hg = gh'$ for some $h'\in H$, we obtain $h = hgg^{-1} = gh'g^{-1}\in ghg^{-1}$. So $gHg^{-1}=H$.  (only if) Let $gHg^{-1}=H$, $g\in G$; if we take an $l\in Hg$, $l = hg$ for some $h\in H$, we note immediately that $l=hg=(gh'g^{-1})g=gh'\in gH$. What should I do to show that $gH\subset Hg$?  I'm also wondering if this mess is correct, and if it is possible a more elegant proof of this fact. What are the grammatical rules determining whether to use "which" or "what"? What are the grammatical rules determining whether to use which or what?  I usually go by intuition. It's "What car are you looking at?" and "What bicycle do you like most?" but "Which song did you sing?" Quite often, I don't know whether to use what or which.  How can I tell? I'm new to latex, I'm thankful if someone can give me an example script to customize TOC that looks like  Chapter 1.XXXXXX                                                            (1-15)  1.1 Introduction 1.2 xxxxxxxx 1.3 xxxxxxxxxx 1.4 xxxxxxx ...............  Chapter 2.xxxxxxx  2.1 xxxxx ......                                                           (16-30)  because the (default) TOC appears to be very large for my book, since it arranges the section name row-by-row. I would like to reload an &lt;iframe&gt; using JavaScript. The best way I found until now was set the iframe’s src attribute to itself, but this isn’t very clean. Any ideas? Linux network troubleshooting and debugging From time to time Linux and Unix users faced with various network problems. Many of these problems are presented here and at some others troubleshooting forums, but they are very concrete and contains a lot of additional technical information, and sometimes it's rather difficult to understand the main point and the real reason of buggy system behavior.   By asking this question, my intention is to start a community wiki page which allows generalizing our network troubleshooting and debugging experience. I hope the Linux and Unix users could easier recognize and solve("divide and conquer") their network problems using this page.   The parent of this page should be  . But here we should focus on troubleshooting the network problems from user- and kernel-space.  I suppose, if you:    Share the information about using some great network diagnostic tool with concrete usage examples and examples of network bugs, which they help to catch. Share the link to the great network tutorial connected with this subject Tell about a general method or recipe which allows to tackle some class of network problems Share information about your tool-set for network debugging and troubleshooting   it would perfectly fits for this topic.    I'll begin from sharing the link to varios  and . Also  seem to have actual information about our subject. And for diving into linux networking we definetely need to visit . Undefined control sequence \documentclass First of all I would like to thank you because you helped me out a lot with all your answers to the already online questions.  Now, here is my problem. I am trying to do a bibliography file with TeXworks but I get this error when I try to compile (pdfLaTeX+MakeIndex+bibTeX):      This is pdfTeX, Version 3.1415926-2.4-1.40.13 (MiKTeX 2.9 64-bit)     entering extended mode     ****PATH here******     ! Undefined control sequence.     l.1 \documentclass         [a4paper,11pt]{report}   I've been reading some similar questions but I don't find the way to fix it (maybe I am missing something). I found one that says that I am trying to compile a LaTeX file using TeX compiler but I don't know how to change the configuration.  This is the .bib fiel content:      \documentclass[a4paper,11pt]{report}     \usepackage[english]{babel}     % faire de l'anglais     \usepackage[latin1]{inputenc}   % accents dans le source     \usepackage[T1]{fontenc}        % accents dans le DVI     \usepackage{url}     \begin{document}      @Article{ref11, author={Danihelka, Jiri and Kencl, Lukas}, journal={Cloud Futures Workshop 2012 }, title={Colaborative 3D Environments over Windows Azure}, year={2012}, month={May}, pages={}, note={}     }      @book{ref12, author = {Lee, Henry and Chuvyrov, Eugene},     title = {Beginning Windows Phone 7 Development},     publisher = {Apress},     volume = {},     number = {},     series = {},     address = {},     edition = {Second edition},     year = {2011},     month = {July},     note = {}     }      @Article{ref13, author={}, journal={}, title={Cloud computing}, year={}, month={}, pages={}, url={\url{http://en.wikipedia.org/wiki/Cloud_computing}}     }      \end{document} Graph nodes coordinates evaluation