Labeling the diagonal/side between $\mathrm{i}$ and $\mathrm{j}$ as $\mathrm{i}+\mathrm{j}$ (reduced if necessary mod $\mathrm{n}$) almost works. The labels for all the lines at a given vertex will be different. But the line between i and $n$ will have label i, the same as one endpoint. However, we are not using the label 2i for the lines from vertex i. So for the line between $i$ and $n$ we use $2 i$ instead of $i+n$. The only points that need checking are (1) whether a line from $\mathrm{i}$ to $\mathrm{n}$ has a label different from $\mathrm{n}$, and (2) whether all the lines at $\mathrm{n}$ have different labels. Both points are ok because $\mathrm{n}$ is odd.