Existence of regular triangle- and tree-free graphs

Construct, or determine the existence of, a (k−1)-regular graph on n₀ vertices that is both triangle-free and T-free for every relevant n₀ and every balanced tree T of order 2k or 2k+1 lacking the matching condition used in the paper.

Background

The extremal constructions require a (k−1)-regular graph on the larger part of a complete bipartite graph that avoids the decomposition-family obstructions associated with the suspension ̂T. Because the relevant members other than K₁,ₖ contain triangles, the authors reduce the construction challenge to finding a (k−1)-regular {K₃,T}-free graph of every required order. They explicitly state that even the existence of such graphs is unknown.