Characterization of boundary forbidden families

Determine the boundary forbidden families for which the inclusion of cycle-maximal graphs in edge-extremal graphs, namely CEX(n,𝓕)⊆EX(n,𝓕) for all sufficiently large n, holds when the extremal number has the boundary form ex(n,𝓕)=t_r(n)+γ_r n+O(1).

Background

The paper establishes the sharp universal linear coefficient γ_r in the Turán remainder for r≥3. Under the strict bound ex(n,𝓕)<t_r(n)+γ_r n for all sufficiently large n, it proves that every cycle-maximal 𝓕-free graph is edge-extremal. It also constructs counterexamples showing that the conclusion fails for some forbidden families at the boundary coefficient.

At the boundary, the leading comparison between cycle counts can vanish. The authors explain that the next-order comparison depends on the edge difference, part sizes, adjacent internal edge pairs, and bounded exceptional components. Consequently, the value of the linear coefficient alone does not determine whether CEX(n,𝓕)⊆EX(n,𝓕), leaving the classification of boundary families unresolved.