Proove that a 3-regular bridgeless graph has perfect matching?
Problem. Use Tutte's 1-factor theorem to prove that every 3-regular graph with no bridges has a 1-factor.  I honestly have no idea how to approach this problem. I have drawn a couple of graphs to gain some insight to no avail. Any hints or tips will be greatly appreciated.
referring to Eqn. 1a) and 1b) in a sentence as eqn. (1) How can I successfully finish a hobby game project in my free time?  How do I motivate myself to keep it up and drive right through to the finish without losing interest or motivation along the way? An example of grey and white note that I mean is, like when you copy something, a small grey background and white lettered bubble appears for a few moments saying 'copied to clipboard'. My phone has some sort of malfunction where it dispays that prompt for several minutes at a time. Very annoying. How do I disable that? I am using a Samsung Galaxy S9. How can I get to Iceland on a cargo ship? Planning a low budget and environment friendly trip to Iceland from Helsinki, Finland next summer for me and my friend. Haven't been able to find relevant information on possibility for cargo travel (from where - eg. Norway, Denmark? - timetables, prices etc). Expected departure from Helsinki around June 18th, return at latest on July 7th 2015. I'm very thankful for any related advice (and first time on stackexchange, so apologies if I didn't formulate my question in a best possible way) How to embed a text file in a .NET assembly? I would like to embed a text file in an assembly so that I can load the text without having to read it from disk, and so that everything I need is contained within the exe. (So that it's more portable)  Is there a way to do this?  I assume something with the resource files?  And if you can, how do you do it and how do you programaticaly load the text into a string? I am a novice user and I am trying to install Oracle Java. The process calls for me to access /usr/local/java, but here is the problem. I get as far as /usr/local, then this is the message I get:  leonard@leonard-MT6452:/usr/local$ dir bin  etc  games  include  java  lib  man  sbin  share  src  leonard@leonard-MT6452:/usr/local$ cd /java/ bash: cd: /java/: No such file or directory  leonard@leonard-MT6452:/usr/local$ rm -r ~/java rm: cannot remove ‘/home/leonard/java’: No such file or directory   Help? Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$? We know that $f(x)=x^4+1$ is a .  My question is:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ has a linear factor over $\mathbb{Q}$?   Edit: Thanks for @Jyrki Lahtonen's answer, I want to do some modifications:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?   Thanks in advance! Best practice when analysing pre-post treatment-control designs How to do binary calculation？ Name for a word whose sound is contrary to its meaning How do I size a UITextView to its content? Is there a good way to adjust the size of a UITextView to conform to its content? Say for instance I have a UITextView that contains one line of text:  "Hello world"   I then add another line of text:  "Goodbye world"   Is there a good way in Cocoa Touch to get the rect that will hold all of the lines in the text view so that I can adjust the parent view accordingly?  As another example, look at the notes' field for events in the Calendar application - note how the cell (and the UITextView it contains) expands to hold all lines of text in the notes' string. What makes a bread either close or open crumbed/textured? What causes the texture of bread to be open, ie with lots of large holes, or close, ie a regular, uniform tender crumb with no large holes? How can I achieve either? How does \fontsize{}{} work? With the package anyfontsize I can use the command \fontsize{}{}. But the second argument seems to be without any effect. I read  that it is related to the line space. It doesn't work.  \documentclass{article} \usepackage{anyfontsize} \begin{document}  {\fontsize{1cm}{1cm}\selectfont First test : I need to put some text here.}  \bigskip  {\fontsize{1cm}{2cm}\selectfont Second test : I need to put some text here --- does it work?}  \end{document}     Any clue? I'm using Emacs with AUCTeX to do all my LaTeX. When it comes to syntax highlighting it seems that all mathematics is by default just orange. Do you know of a mode or package that adds some more fine grained syntax highlighting? For example all commands (\foo), matching braces, sub- and superscripts, etc. should get a different color. Is the Doctor Who episode 'Blink' the first use of the phrase "timey-wimey"? Case-insensitive string comparison in C++ PCA and the train/test split I have a dataset for which I have multiple sets of binary labels. For each set of labels, I train a classifier, evaluating it by cross-validation. I want to reduce dimensionality using principal component analysis (PCA). My question is:  Is it possible to do the PCA once for the whole dataset and then use the new dataset of lower dimensionality for cross-validation as described above? Or do I need to do a separate PCA for every training set (which would mean doing a separate PCA for every classifier and for every cross-validation fold)?  On one hand, the PCA does not make any use of the labels. On the other hand, it does use the test data to do the transformation, so I am afraid it could bias the results.  I should mention that in addition to saving me some work, doing the PCA once on the whole dataset would allow me to visualize the dataset for all label sets at once. If I have a different PCA for each label set, I would need to visualize each label set separately. I want to know how to get the X and Y position of HTML elements such as img and div in JavaScript relative to the browser window. Method to create a live USB disk (with persistence) which actually works? I've tried so many combinations - all I want is a fast Ubuntu Live USB disk with persistence that works. What I've tried so far:   Using unetbootin on my Mac, choosing some "Space used to preserve files across reboots (Ubuntu only)".   Result: Doesn't boot at all.  Using unetbootin on my Mac, choosing no such space.  Result: Boots fine and very fast (doesn't even show the annoying dialog "Try ubuntu/install ubuntu" at startup!), but has no persistence. (So I have to choose my keyboard layout, install programs I need, etc. every time)  Using Startup Disk Creator from my Ubuntu 11.04 installation.  Result: I cannot choose "reserved extra space" at all. It's greyed out. I know I know, there was bug , and many people say that the workaround from  works for them. But it doesn't work with Ubuntu 11.04 anymore (see also ).  Using Startup Disk Creator from my Kubuntu 10.10 installation, gnome version (usb-creator-gtk).   Result: I can select the "reserved extra space" option, but installation doesn't work (fails when installing the bootloader).  Using Startup Disk Creator from my Kubuntu 10.10 installation, kde version (usb-creator-kde).   Result: Actually works: Reserved space can be selected, installation works. But after booting, I get the screen "Try ubuntu/install ubuntu". Clicking on "Try ubuntu" takes me to the desktop - but only after waiting 5 minutes (every time!). It also doesn't seem to keep my keyboard settings.    I know, that instead of creating a startup disk, I could install a full ubuntu system on the USB stick. But as far as I know, this isn't very good, because /tmp and similar directories will then also be on the USB disk, which is a) slow, and b) not very good for the USB disk due to frequent overwriting.  I'm frustrated. So has anybody managed to create a working Ubuntu 11.04 Live USB disk with persistence? I am writing a few rather large pseudocode functions using algorithm and algorithmic, i.e. my code looks something like:  \begin{algorithm}                      \caption{Coolest Algorithm ever}           \label{findme}                           \begin{algorithmic} [1]                   % enter the algorithmic environment \REQUIRE    Here are a few variables \\             variable 2 \\             variable 3  \ENSURE     this is the output \\             output 2  \STATE $some cool code here$ \STATE $some cool code here$ \STATE $some cool code here$ \STATE $some cool code here$ \end{algorithmic} \end{algorithm}   Unfortunately this code does not fit on one page and I am definitely not able to split it into some subfunctions or so (I already did that as far as possible).  So now I need to split up the algorithm into two pages, but I have no idea how.  The easiest way I can imagine is just to start a new algorithm on the new page, but this time letting the line counter start at line 35 or so instead of 1. But how can I do that?  Please not that there is already a similar question here on this site, but it deals with the package algorithmicx.