If $G$ is a graph, let $G'$ denote the graph obtained by doubling each edge of $G$. How can I show that $\chi'(G')=2\chi'(G)$?
I am considering the two cases when $G$ is a complete graph $K_n$ with $n$ odd or even. If $n$ is odd then, I know that $\chi'(G)=\chi'(K_n)=n$ and $n+1$ if $n$ is even. I also showed in a previous problem that $\chi'(G')\leq 2\chi'(G)$ giving an upper bound. So, I need to give a lower bound for the chromatic index. If each edge is doubled, then $\Delta(G')=2\Delta(G)$. This is how far I got. Thanks in advance!
Take $C_5$, the cycle graph of length 5.
We have $\chi'(C_5) = 3$.
Let $C_5'$ be the graph obtained after doubling the edges of $C_5$.
You should be able to show that $\chi'(C_5') = 5 < 2 \cdot 3$.