Can somebody, please, explain how to prove this theorem: If every vertex of a planar graph has an even degree, it’s 2-colourable. I understand that we can make a dual graph out of original. The number of vertices will be the same as number of faces in the original graph. The number of edges will be the same as in the original graph. But what should I do next to prove the theorem?
My current idea is to use the dual graph, and prove that that is a bipartite graph, and so proving the statement. But I can't seem to figure out how even degree for all verticies means that it is bipartite.  Any help is appreciated. Thanks!
If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $. I solved it using AM, GM inequalities and reached to $[(1+a)(1+b)(1+c)]^7 &gt; 2^{21}(abc)^\frac72 $ please help how to get $7^7(abc)^4$ in the inequality. Unity Tweak tool fails to start with the error "com.canonical.unity.webapps schema is missing" com.canonical.unity.webapps   In order to work properly, Unity Tweak Tool recommends you install the necessary packages What is the most effective way for float and double comparison? What would be the most efficient way to compare two double or two float values?  Simply doing this is not correct:  bool CompareDoubles1 (double A, double B) {    return A == B; }   But something like:  bool CompareDoubles2 (double A, double B)  {    diff = A - B;    return (diff &lt; EPSILON) &amp;&amp; (-diff &lt; EPSILON); }   Seems to waste processing.  Does anyone know a smarter float comparer? Can Tamron 16-300mm Macro Lens fit to Canon EOS 600D? I use Canon EOS 600D and I want a lens with a larger range. I found out Tamron 16-300 f/3.5-6.3 Di II VC PZD Macro Lens, I added it into my wishlist. I did some research on it and I'm scared that it couldn't fit to my Canon EOS 600D.  I'm still a beginner... I'm confused with EF, EF-S and APS-C/DX. Can someone please explain it to me? Why is methane considered an important greenhouse gas? Methane’s absorption bands are centered at 3.2 and 7.2 microns -- far off the peak of the Planck spectrum for a 290 K blackbody near 16 microns.  Moreover, its absorption bands overlap with the water absorption region.  So why is methane considered a stronger greenhouse gas than carbon dioxide, which has an absorption band at 15 microns? How do I make sure that values from MySQL keep their type in PHP? Does $\sum_{k=1}^{\infty}\ln(\frac{k}{k+1})$ converge/diverges?? Does this series converge or diverge? $$\sum_{k=1}^{\infty}\ln(\frac{k}{k+1})$$   my thought is that, I can break it down to $$\sum_{k=1}^{\infty}\ln(k) - \sum_{k=1}^{\infty}\ln(k+1)$$ then maybe using comparison test or something? But I don't know exactly how to prove whether this series converges or diverges.  Any help would be appreciated! Identifying a thin, black, worm-like creature with a hammer-shaped head I came across the creature shown in the photo below yesterday evening while walking the dogs. At first, I thought it was an earthworm, but a flashlight revealed it to be something I had never seen before.    It is approximately 12 inches (300 mm) in length and about 1/3 of an inch (10 mm) in diameter at its maximum. It is pitch black and very shiny—like piano black. It moves very slowly, using a snake-like motion. It had a very strange head, like a hammerhead shark.  I live in Durban, South Africa, if that helps to narrow down the possibilities.  Can anyone help in identifying this creature? Parameterize an SQL IN clause How do I parameterize a query containing an IN clause with a variable number of arguments, like this one?  SELECT * FROM Tags  WHERE Name IN ('ruby','rails','scruffy','rubyonrails') ORDER BY Count DESC   In this query, the number of arguments could be anywhere from 1 to 5.  I would prefer not to use a dedicated stored procedure for this (or XML), but if there is some elegant way specific to , I am open to that. In a comment on this , someone said that they weren't sure what functools.wraps was doing. So, I'm asking this question so that there will be a record of it on StackOverflow for future reference: what does functools.wraps do, exactly? I remember I saw a apt mirror somewhere containing older distributions but I cannot find it anymore. Does anybody know where to find it? Comma separated values in one column - SQL SERVER Which one is correct? (Maybe both are correct.)     He passed away on the morning of March 5.   Or      He passed away in the morning of March 5. How can I search for files in all sub-directories using the shell and then copy those files? How can I find just the *.mp3 files with a recursive command like ls -R *.mp3 in a directory that contains several sub-directories and at the end copy these files in a directory that I choose.  Thanks for your support. Using the same code behind file for multiple .aspx pages "did shoot" vs "shot" Why does the Sun's (or other stars') nuclear reaction not use up all its "fuel" immediately? The temperature and pressure everywhere inside the Sun reach the critical point to start nuclear reactions - there is no reason for it to take such a long time to complete the reaction process.  Just like a nuclear bomb will complete all the reaction within $10^{-6}$seconds.  Why does most of the hydrogen of the Sun still not react even though it reaches the critical point, and why take stars billions of years to run out of fuel? Unhandled event loop exception in plugin org.eclipse.ui In our campaign we have several characters who never remove their armor. They wear it while adventuring, sleeping, swimming, bathing, etc.  Are there any penalties for:   Wearing armor while sleeping? Wearing armor while swimming?   If there are penalties, do they change based on the category of armor (light, medium, heavy)?  I am primarily looking for RAW penalties specified for armor, but I'm also open to applications of other rules (eg - higher DC for swimming in heavy armor, disadvantage during some situations). Persistent Dummy Output