Question on graphs

Question

Given explanation:

I couldn't understand why S1 is false. Please explain

Answer 1

both are true

1st one :for any graph with n vertices maxdegree of vertexis n-1

so accto pigeonhole principle atleast  two vertices  have same degree

2nd one:

there are even no of odd vertices in any grpah

here  3 vertices of odd degree given not possible

Comments

As we have odd number of vertices with degree odd i.e (3 vertices with degree 5 ) then how this grraph is possible

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As we have odd number of vertices with degree odd i.e (3 vertices with degree 5 ) then how this graph is possible

they said only, 10 vertices, in remaining  3 vertices, we can make one vertex with odd degree.

11,8,5,5,5,4,4,4,4,4,4,4,0 ===> 13 vertices, three vertices have degree 5 and 7 vertices have degree 4

the summation of degrees = 11+8+15+28+0 = 62 = 2*31 = 2 |E|

then S2 is false, right?

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9,7,5,5,5,4,4,4,4,4,4,4,3 ---- one more example