Let $G$ be an outerplanar graph. Is chromatic index of $G$ at most $4$?
I proved that if $G$ is outerplanar then $G$ has chromatic number at most $4$ which can be proved easily using $4$ colour theorem.
However I am stuck here. Is there any relation between chromatic number and chromatic index of a graph?
Also using Vizing's Theorem we know that chromatic index of any graph $G$ is either equal to the maximum degree or maximum degree+1 of a vertex.
Can someone please help me out to prove or disprove this?
There is no connection between these two numbers for outerplanar graphs.
Chromatic number Using the 4CC to prove that the chromatic number of a simple outerplanar graph is at most $4$ is like using a sledgehammer to crack a nut. Furthermore the chromatic number can be proved to be at most $3$.
https://en.wikipedia.org/wiki/Outerplanar_graph#:~:text=According%20to%20Vizing's%20theorem%2C%20the,one%20plus%20the%20maximum%20degree.
Chromatic index
In the comments it is pointed out that there is no bound on this number for simple outerplanar graphs.