Let $G$ be a connected planar graph with a planar embedding where every face boundary is a cycle of even length. Prove that $G$ is bipartite.   It is quite easy to prove the converse, but how to do this? My line of thought is to assume an odd cycle and show a contradiction, but I don't know where to head? This is homework, so please give hints only, not a full solution.
Let $G$ be a connected planar graph with a planar embedding where every face boundary is a cycle of even length. Prove that $G$ is bipartite.  Any hints/tips will be greatly appreciated.
How to minimize $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + |z_3 - z_4|^2$? If $z_1,z_2,z_3,z_4 \in \mathbb{C}$ satisfy $z_1 + z_2 + z_3 + z_4  =  0$ and $|z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 = 1$, then the least value of $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + |z_3 -  z_4|^2$ is $2$.   I solved this by taking two of the values to be $\frac{1}{2}$ and other two as $-\frac{1}{2}$. But I couldn't prove this in an efficient manner. Clicking on tags in global search results is broken List of PDOStatement::bindParam data_type parameters A word for "intimate friends" without any kind of romance? I'm looking for a word to describe that a particular situation where two people (a man and a woman) are probably the best of friends but have no romantic inclinations towards each other. They may invade each others thoughts periodically but aren't really in a relationship per se. What would be a good word to describe this?  I'm aware of "platonic" but am wondering if there is another word? I need a list of integers from 1 to x where x is set by the user.  I could build it with a for loop eg assuming x is an integer set previously:  List&lt;int&gt; iList = new List&lt;int&gt;(); for (int i = 1; i &lt;= x; i++) {     iList.Add(i); }   This seems dumb, surely there's a more elegant way to do this, something like the Why do I not see evidence of discrete energy in the spectrum? The electromagnet spectrum looks to be continuous, yet energy is discrete. Shouldn't one see evidence of the electron's "quantum leap"? What's going on?     Moderator, please size my photo. Thank you. Remove votes to close question X as a duplicate of Y if question Y is closed as a duplicate of X I just happened across , which has one close vote suggesting that it be closed as an exact duplicate of .  This suggestion is perfectly correct: these are exactly identical questions asked by the same user.  However, there's not much point in closing question A as a duplicate of B, since question B has already been closed as a duplicate of A.  Yet the close vote remains, doing nothing but wasting reviewers' time.  This is not the first time I've seen this happen, so I'd like to suggest that the Stack Exchange software should do something about it.  In particular, I'd like to suggest the following feature:     When a question Y is closed as an exact duplicate of another question X, any votes or flags to close X as a duplicate of Y should be automatically discarded.   Actually, I'm kind of surprised that the SE software doesn't do that already.  If anyone else thinks that this constitutes an actual bug, feel free to retag it. Centering vertically in table cell I'm new with LaTeX and I'm trying to make a table with the elements centered vertically and horizontally. I've searched for this question, it seems like most people were trying to do things a little more complicated than this, resulting in complicated answers I couldn't understand. The table below is already centered horizontally, but the elements are aligned to the top of the cells instead of the center. How do I fix this?      \begin{tabular}{*{5}{|c}|}         \hline         $p$ &amp; $q$ &amp; $p \land q$ &amp; $p \lor q$ &amp; $p \to q$\\         \hline         T &amp; T &amp; T &amp; T &amp; T\\         \hline         T &amp; F &amp; F &amp; T &amp; F\\         \hline         F &amp; T &amp; F &amp; T &amp; T\\         \hline         F &amp; F &amp; F &amp; F &amp; T\\         \hline     \end{tabular} Is soaking beans 24 hours unrefrigerated safe? Changing order of features in shapefile Record Animation In BGE Correct usage of ‘on’, ‘at’ and ‘in’ How to Backup and Restore Android OS without Rooting I want to find a way to restore my phone in case the process of rooting my phone or dual booting a custom rom breaks it. However, most of the ones I've looked at require the phone to be rooted before installing. I know I can create a nandroid backup through system settings, but I don't know how to recover that without rooting my device, or if there are better options. I'm not terribly concerned for any personal data on my phone (I can backup my pictures pretty easily), but I mostly just want to be able reinstall Android OS or reset to factory default in case my device breaks. Thanks! Edit: I am going to ask a new question (one different enough to justify not simply modifying this one) with the new info I learned from the possible duplicate. Skipping acquire of configured file 'contrib/binary-i386/Packages' as repository ... doesn't support architecture 'i386' I installed Ubuntu 18.04 and then while installing Virtualbox i get the 'i386' error message after I did:  $ sudo apt-get update Get:1 http://nl.archive.ubuntu.com/ubuntu bionic InRelease [242 kB] Hit:2 http://security.ubuntu.com/ubuntu bionic-security InRelease                                                                 Hit:3 http://ppa.launchpad.net/danielrichter2007/grub-customizer/ubuntu bionic InRelease                                          Hit:4 http://nl.archive.ubuntu.com/ubuntu bionic-updates InRelease Hit:5 http://nl.archive.ubuntu.com/ubuntu bionic-backports InRelease Get:6 http://download.virtualbox.org/virtualbox/debian bionic InRelease [4429 B] Get:7 http://download.virtualbox.org/virtualbox/debian bionic/contrib amd64 Packages [1426 B] Fetched 248 kB in 1s (209 kB/s)      Reading package lists... Done N: Skipping acquire of configured file 'contrib/binary-i386/Packages' as repository 'http://download.virtualbox.org/virtualbox/debian bionic InRelease' doesn't support architecture 'i386'   I found several answers on Askubuntu for the 'i386' message but I don't know how to apply this to my situation How to pronounce '-ing' followed by a vowel I'm getting into English recently and I'm a little confused by the way people pronounce a word that starts in a vowel right after a word ending in -ing.  For example:      You have to bring it up now?   I don't know if I should pronounce it as:   bring /ŋit/ brin' /nit/ bring /git/   Another example would be "I'm coming out".  Likewise, what about the ending -ang? For example:      We can just hang out and have a good time.    And to round it all up, what about a combination of the two? For example,      He got sick of hanging around waiting for you and went home.   I've been asking myself these questions for weeks. The question asks it all really. When referring to a master's degree, do you use an apostrophe or not? That is, is it "a master's" or "a masters"? Why is no root pass needed to access MariaDB after upgrade to Ubuntu 16.04? I’ve had this issue for months but in the last couple of days it’s got very very bad.  When my phone runs out of battery it needs like 2–3 days to turn back on. At the beginning of this problem the phone would need about 1h. Every time when my phone goes off again the time to switching on goes up.  What does this all look like? Well, when my phone died I put it on the charger, then the logo appears and goes off. When it goes off, a black screen pops out with a little white line on in it, then it turns on back to Apple logo. It keeps doing this until it turns on which is 1 or 2 even now 3 days.  I tried to restore it and also update it but nothing happens, literally nothing happened. Sometimes when I am debugging code in Eclipse it happens that although I can see and inspect class member variables without any difficulty I am unable to inspect the values of variables declared locally within functions. As an aside, any parameters to the current function lose their 'real' names and instead one sees their values listed in the Variables window as arg0, arg1, arg2, etc but at least the values are visible.  This is occurring at present in relation to classes defined within the core JDK. I have verified that the installed and current JRE is a JDK.  Is anybody able to shed some light on this behaviour? How can I use Web.debug.config in the built-in visual studio debugger server?