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 Feb 1 awarded Popular Question May 30 comment Can any planar graph have 4 vertices and 4 regions? Actually I am confused at that point because the book (Grimaldi, Discrete and Combinatorial Mathematics) I read considers the region outside as infinite region and takes it into account. May 30 comment Can any planar graph have 4 vertices and 4 regions? @kyticka in short yes, here you can read en.wikipedia.org/wiki/Planar_graph May 30 comment Can any planar graph have 4 vertices and 4 regions? See the second image from top. I'm considering this represenation. rip94550.wordpress.com/2008/11/30/… May 30 revised Given $G = (V,E)$, a planar, connected graph with cycles, Prove: $|E| \leq \frac{s}{s-2}(|V|-2)$. $s$ is the length of smallest cycle added 426 characters in body May 30 comment Given $G = (V,E)$, a planar, connected graph with cycles, Prove: $|E| \leq \frac{s}{s-2}(|V|-2)$. $s$ is the length of smallest cycle Ok, so observe that $2e \geq sr$, $r$ is region in your notation it is $f$, face. May 30 answered Given $G = (V,E)$, a planar, connected graph with cycles, Prove: $|E| \leq \frac{s}{s-2}(|V|-2)$. $s$ is the length of smallest cycle May 30 asked Can any planar graph have 4 vertices and 4 regions? Jan 13 awarded Commentator Jan 5 comment How many elements of order d? @DonAntonio thank you for your feedback! Can you mention me the way you thought to find the answer, I'm interested in different solution Jan 5 comment How do I find this limit? @BabakSorouh the 2 answers below are actually what I meant, seems like working Jan 5 comment How do I find this limit? you can start with taking the ln of limit Jan 5 asked How many elements of order d? Jan 2 awarded Enthusiast Dec 21 awarded Supporter Dec 21 awarded Student Dec 21 asked How to evaluate $\int_0^{2\pi} \frac{d\theta}{A+B\cos\theta}$? Dec 21 comment How do I find the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$? can it be possible that the trivial one should not be taken into account, otherwise I dont see any explanation, I'm quite what I've explained. Dec 18 answered How do I find the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$? Dec 6 answered On Ceva's Theorem?