Existence of regular triangle- and tree-free graphs of every required order

Construct, or determine the existence of, a (k−1)-regular graph on n₀ vertices that is both triangle-free and T-free for every n₀, in order to establish equality in the Turán bound for every n for the suspensions of the relevant balanced trees.

Background

The extremal constructions embed a (k−1)-regular graph into one part of a complete bipartite graph. To obtain equality for every n, the paper identifies the need for such a regular graph on the required part size n₀. Since the other members of the decomposition family besides K₁,k contain triangles, the authors reduce this construction challenge to finding a {K₃,T}-free (k−1)-regular graph. They explicitly state that existence for every n₀ is unknown.