Is the sequence $(4,4,4,2,2)$ graphical? Conclusion: it seems it

shouldn't be$,$ because there are too many vertices of degree $4$ and not enough

other vertices of high enough degree to absorb the edges these necessitate.

One theme of this course is that we will be more interested in how we can

adapt the methods of proof to prove related results, and when strengthenings

of results are possible.

If we look at the proof of Havel$-$Hakimi, all that we needed for the proof to

work was that the vertex $v_j$ had a neighbour that $v_i$ didn't have. Initially, we

made an assumption that $d(v_j)>d(v_i),$ which guaranteed this for us$.$ However,

we soften remove this assumption, for if $d(v_j)=d(v_i)$ and $v_{1}in\frac{v_i}{\frac{?}{?}}(v_j),$

then this also guarantees the existence of such a $u.$

Exercise $1\mathrm{.}25.$ Adapt the statement and proof of Havel$-$Hakimi in light of this

observation.