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 May30 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. May30 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 May30 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/… May30 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 May30 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. May30 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 May30 asked Can any planar graph have 4 vertices and 4 regions? Jan13 awarded Commentator Jan5 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 Jan5 comment How do I find this limit? @BabakSorouh the 2 answers below are actually what I meant, seems like working Jan5 comment How do I find this limit? you can start with taking the ln of limit Jan5 asked How many elements of order d? Jan2 awarded Enthusiast Dec21 awarded Supporter Dec21 awarded Student Dec21 asked How to evaluate $\int_0^{2\pi} \frac{d\theta}{A+B\cos\theta}$? Dec21 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. Dec18 answered How do I find the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$? Dec6 answered On Ceva's Theorem? Dec6 revised Decomposing a subgroup I've needed to first prove this on my own before sharing.