$(3)($i$)$ If a graph with $7$ vertices has degrees $6,6,4,3,3,3,d,$ what is the smallest value that $d$ could have? Draw an example of a graph for which $d$ takes this value.

$($ii$)$ What is the least possible number of edges in a connected graph with $6$ vertices in which each edge is in at least two cycles? Draw an example of such a graph with this number of edges.

$($iii$)$ Let $n4$ be an even integer. What is the greatest possible number of edges in a graph with $n$ vertices in which each vertex has even degree? $($Give your answer as an expression in terms of $n.)$ For $n=6,$ draw an example of such a graph with this number of edges.

$[$Answers only required $($this includes a picture for each part$).]$

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