$\color{red}{\text{Detailed Video Solution, with Concept:}}$ https://youtu.be/42tvGJDgu-I
Easiest & Correct way to solve it:
First understand the given graph $G$:
It has maximum degree $\Delta = 7,$ degree sequence $7,5,5,5,4,4,4,4$ (Vertex $s$ has degree $7$, three vertices have degree $5$ and remaining four vertices have degree $4$)
Now, Some Important Theorems about Edge Coloring:
1. In the Edge Coloring of a graph $G:$ The least number of colors needed to color the edges of $G$ so that any two edges that share a vertex have different colors, is called Chromatic Index of $G$ and it is denoted by $\chi’(G).$
2. For any simple graph, $\chi’ \geq \Delta $ (Easy Proof HERE)
For the given graph, $\Delta = 7,$ So, $\chi’ \geq 7.$
3. Vizing’s Theorem: For any simple graph, $\chi’ = \Delta$ OR $\Delta + 1.$
For the given graph, $\Delta = 7,$ So, $\chi’ = 7$ OR $8.$
Note that Graphs with $\chi’ = \Delta $ are called Class-1 graphs, and Graphs with $\chi’ = \Delta + 1$ are called Class-2 graphs.
4. MOST Interesting & Useful Result by Vizing:
Vizing proved that Every graph with a unique vertex of maximum degree must be Class-1. More specifically, he showed that every class two graph must have at least three vertices of maximum degree.
So, if a graph $G$ has $\leq 2$ vertices of degree $\Delta,$ then $G$ is Class-1 graph, Hence, $\chi’(G) = \Delta.$
For the given graph, there is Unique vertex of maximum degree $\Delta = 7,$ So, given graph is Class-1, Hence, $\chi’(G) = 7.$
Detailed Video Solution, with Concept: https://youtu.be/42tvGJDgu-I
Sources:
Source of Point 4: https://math.uchicago.edu/~may/REU2015/REUPapers/Green.pdf (Page 3, Paragraph 2)
https://core.ac.uk/download/pdf/82035825.pdf