Show that if $G$ is Hamiltonian, then $L(G)$ is Hamiltonian. Show that if $G$ is Hamiltonian, then $L(G)$ is Hamiltonian. Attempt: For any graph, we can relabel the vertices.  For a graph $G$, then we must have a Hamiltonian cycle $(1, 2, \ldots , n)$. Now, I think it is just a matter of guaranteeing that in $L(G)$ there is a cycle $((1,2), (2,3), \ldots )$.
Proving that if $G$ has a Hamiltonian circuit, then the line graph $L(G)$ also has a Hamiltonian Circuit I have a graph G = (V, E). I know that the line graph L(G) of G is defined as follows: each vertex in L(G) corresponds to an edge in G, and two vertices are connected by an edge in L(G) if and only if the corresponding edges in G are adjacent.  Now I am trying to (a) prove that if G has a Hamiltonian circuit, then L(G) also has a Hamiltonian circuit, and (b) give an example of a graph G such that G does not have a Hamiltonian circuit but L(G) does.  However, I'm extremely stuck on how to do this, and any help would be greatly appreciated. Thanks so much!
Cooperative game vs Multiplayer game Can any one tell what is difference between cooperative game and multiplayer game? Seems that I just proved $2=4$. How to make two people play at the same time? My brother owns Skyrim on steam. We use Steam Family Share, but we were wondering how both of us could play it at the same time? Typesetting long division with tabular and overlining repeating part How do I delete a board in Trello? How can I get the list of files in a directory using C or C++? How can I determine the list of files in a directory from inside my C or C++ code?  I'm not allowed to execute the ls command and parse the results from within my program. Cardinality of the quotient ring $\mathbb{Z}[x]/(x^2-3,2x+4)$ This problem is from a practice exam I was working on.     What is the cardinality of the quotient $\mathbb{Z}[x]/(x^2-3,2x+4)$ ?   Thoughts. If I find a ring that is easier to handle then this then I can go from there. So I think this is isomorphic to $\mathbb{Z}_2[x]/(x^2-3,x+2)$. And then I am stuck.  Am I correct so far? What should I be looking to do from here?  Thanks for your time and your answers. In the magical world (in Harry Potter) can ghosts use magic?  Can magic be used on ghosts?  Or, once they become ghosts, can they interact with magic at all? Why don't we treat rubbish the same as spam? Spam gets fairly heavily punished - 100 reputation penalty, if I recall correctly, and the content itself gets hidden in the revision history.  Now if we just have some rubbish:     dfajiojaifojadiofjadhigaowkokaomdiovnuiyhioqejgioqejgio   I'm not talking about someone posting a question or comment as an answer (i.e. possibly just not knowing the rules or being frustrated with not having the required privileges), or someone making some really horrible attempt at answering the question, I'm talking about an answer that clearly shows absolutely no observation of any content on the page.  This, in my opinion, causes a similar amount of harm to the site - I'm inclined to say the fact that someone posts spam isn't as important as the fact that they're posting something that needs to be deleted, since we're likely going to delete it pretty quickly, and they're (likely) behaving maliciously, or am I wrong?  Why don't we treat this equally harshly, or why do we treat spam as harshly as we do?    Side notes:  I'm sure I've seen a few times where users get punished for posting spam when their link is likely actually relevant to the question and useful (should I flag these? They are still link-only answers, so they shouldn't really come back, and often I'm just not sure enough). While false positives are bad, it seems worse that we're treating worse behaviour better.  I'm assuming that's not what "abusive" in this reason is referring to: (I'm also not sure, off the top of my head, how this one gets treated)     It is offensive, abusive, or hate speech How do you prove the second law of thermodynamics from statistical mechanics? How do you prove the  from statistical mechanics? To prove entropy will only increase with time? How to prove? Please guide. Does each season of Game of Thrones roughly equal a book? Does each current Game of Thrones season roughly equal one A Song of Ice and Fire book? And do we know which future seasons will cover which books?  I am planning to read the first book after the TV show has finished depicting the events in the first book. I would like to know if after each TV season ends I can expect to be able to read a book covering the events depicted on screen? JSON.parse: unexpected character Effects of continuing after a total party wipe? I've had a couple of encounters so far where I stumbled into enemies that were higher level than expected and gotten wiped out. At the game over screen, I chose the option "Continue". After doing so, I didn't notice any negative consequences. I was plopped back just a few metres from the area I died, and was free to continue on as if nothing had happened. Am I missing something? Is there some negative effect from a party wipe? Is there evidence of dark matter in our galaxy? Is there evidence of dark matter in our galaxy?  How can we measure this, say, how many percent of the center of our galaxy is dark matter?  I did not find the answer in the question Does League of Legends have a tournament that is recognized as the 'Most Important'? Deleting another user's directories from my own I am a non-root user, and have made a directory into which other users in my group can write. The directory is setgid, so files and directories within it have the same group.  I can delete files placed into this directory, but if a user creates a subdirectory with files in it, I can't seem to delete those. Is there something special I can do (other than, say, bothering the user in question or the sysadmin about it) to get rid of this subdirectory?  EDIT: Inevitably, of course, someone asks "why?"  The shared folder in question is a mercurial repository. We can't use a shared-account ssh-key-based server for this like mercurial-server, and we can't just host it externally for various reasons. But that part works fine. We use Redmine for project management, but it does not yet read the repo index for file contents - it reads the working directory. So the working directory must be kept up-to-date. crontab has been disabled by the sysadmin, so that's out. So I naïvely put a changeset hook in the shared repo that performed an "hg update". This seemed to work fine initially, until, of course, someone pushed a commit with a new directory. While I had vague thoughts of trying to make a setuid script for the hook, I'll probably just consider safer avenues. If I have bought 3 tickets for the same flight, can I bring three times the cabin allowance allowed per ticket?  The way I see it I am doing the airline a favor  Related: How to urlencode data for curl command? I was really happy when I heard that HoTD would be continued. And even happier when the first new chapter actually came. but now after several months of waiting there still has been no other chapter. So my question is, has HoTD gone on hiatus again ? Salesforce lightning get current users username How I would go about getting the current users username or first and last name?  I have been looking at this for the last couple of hours - I was just wondering how I would go about getting their name and then displaying it on the page.  For example the current user is 'John Smith' and in the lightning component I want to output John Smith's name like:     Welcome to our community John Smith!