Referring to a past question of mine that can be found here:
How to demonstrate that there would exist a vertex with degree less than $ 5$?
This is the work that I have come up with following the guidance kindly provided by other members.
Working:
$$\text{Assume that every vertex of the graph have a degree of at least six.}\\ \text{ Using} ∑_{v∈V}deg(v)=2e \text{ for all graphs,}\\ \text{Since } deg(v)≥6 \text{ for all } v∈V \text{ according to the assumption, we have }\\ 2e=∑_{v∈V}deg(v)≥6n \\ \text{would imply } e \geq 3n \\ \text{ yet, according to Euler's theorem } e \leq 3n -6 \\ \text{therefore implying that there is a contradiction.}\\ \text{Now, it can be concluded that the initial assumption cannot be true, and S has at least one vertex with a degree at most five}$$
I was wondering if anyone would be wiling to check my work and provide any correction/advice on how to improve it?
My main issue is that I do not know if this working would be detailed enough.
I do not know if this is allowed on the site.
If not, just let me know and I will delete the question.
Thank you.
