Open Problem Garden

Weak pentagon problem ★★

Author(s): Samal

Conjecture If $G$ is a cubic graph not containing a triangle, then it is possible to color the edges of $G$ by five colors, so that the complement of every color class is a bipartite graph.

Keywords: Clebsch graph; cut-continuous mapping; edge-coloring; homomorphism; pentagon

Posted by Robert Samal
updated July 13th, 2007

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Conjecture Let $f(n) = 3n+1$ if $n$ is odd and $\frac{n}{2}$ if $n$ is even. Let $f(1) = 1$ . Assume we start with some number $n$ and repeatedly take the $f$ of the current number. Prove that no matter what the initial number is we eventually reach $1$ .