New bounds on maximal linkless graphs

We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $n\ge 10$, and another family on $n$ vertices and $m< \frac{25n}{12}-\frac{1}{4}$ edges for all $n\ge 13$. The latter significantly improves the lowest edge-to-vertex ratio for any previously known infinite family. We construct a family of graphs showing that the class of maximal linklessly embeddable graphs differs from the class of graphs that are maximal without a $K_6$ minor studied by L. Jorgensen. We give necessary and sufficient conditions for when the clique sum of two maximal linklessly embeddable graphs over $K_2$, $K_3$, or $K_4$ is a maximal linklessly embeddable graph, and use these results to prove our constructions yield maximal linklessly embeddable graphs.