Existence of suitable regular triangle- and tree-free graphs

Construct, or determine the existence of, a (k−1)-regular graph on n0 vertices that is free of every member of Ŝ ∪ {T}, equivalently in the discussed setting a {K3,T}-free (k−1)-regular graph, for every required value of n0.

Background

The extremal constructions in Section 2 embed a (k−1)-regular graph into one part of a complete bipartite graph. To establish equality in the bound f(n,k) for every n, the embedded graph must be Ŝ ∪ {T}-free, where Ŝ is derived from the decomposition family of the suspension Ť.

The paper provides such constructions for specific orders using disjoint unions of complete bipartite graphs and related components, but does not establish existence for all n0. The authors identify this construction problem as unresolved.