There is a graph with 40 vertices. It is known that any edge has at least one endpoint, on which no more than four other edges are incident. What is the maximum number of edges that this graph can have? No multiple edges or loops are allowed.

mxty42ued 2022-11-17 Answered
Maximum number of edges in a graph satisfying conditions
There is a graph with 40 vertices. It is known that any edge has at least one endpoint, on which no more than four other edges are incident. What is the maximum number of edges that this graph can have? No multiple edges or loops are allowed.
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Answers (1)

Nigerkamg5
Answered 2022-11-18 Author has 20 answers
Step 1
For a vertex v with a neighbours of degree 5 and b neihgbours of degree > 5 we define f(v) as a / 2 + b.
Notice m = v | d ( v ) 5 n f ( v ).
Step 2
Let k be the number of vertices of degree 5.
If k 35 then clearly m 35 5 = 175.
If k = 35 + l with k { 1 , , 5 } then each summand is at most 5 l 2 so v | d ( v ) 5 n f ( v ) ( 35 + l ) ( 5 l 2 ) < 175.
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