Let G be a graph not necessarily simple with 6 vertices and

Let G be a graph (not necessarily simple) with 6 vertices and 10 edges such that every
vertex of G has degree 1, 3, or 5. If the number of vertices of degree 3 is one more that
the number of vertices of degree 5, how many vertices of each degree does G have?

Solution

let assume Number of vertices with degree 5 = x

so Number of vertices with degree 3 = x+1

Number of vertices with degree 1 = 6-(x+x+1) = 5-2x

We know that

Sum of total degrees =2*(Number of edges)

=> 5x + 3(x+1) + 1*(5-2x) = 20

=> 8x +3+5-2x =20

=> 6x = 12

=> x =2

So

Number of vertices with degree 5 = 2

so Number of vertices with degree 3 = 3

Number of vertices with degree 1 = 6-(x+x+1) = 1

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