Phase transition for induced-K1,r+1-free graphs

Prove or disprove that, for every integer r≥3, the class of induced-$K_{1,r+1}$-free graphs with n vertices and m∼γ\binom{n}{2} edges exhibits a phase transition in γ∈(0,1), with one phase having almost all graphs co-r-partite and the other having almost all graphs equal to the disjoint union of a co-r-partite graph and a sparse graph.

Background

The paper establishes an analogous two-phase structural description for induced-K1,3K_{1,3}-free, or claw-free, graphs: above the critical density, almost every graph is co-bipartite, while below it, almost every graph is the disjoint union of a co-bipartite graph and a sparse graph.

The conjecture asks whether this behavior extends to induced-K1,r+1K_{1,r+1}-free graphs for all r≥3, replacing co-bipartite graphs by co-r-partite graphs. The authors note that asymptotics for regular induced-K1,r+1K_{1,r+1}-free graphs may be useful for proving it.

References

For all $r3$, the class $(n,m)$ of induced-$K_{1,r+1}$-free graphs on $n$ vertices and $m\sim\gamma\binom{n}{2}$ edges exhibits a phase transition in the edge density $\gamma\in(0,1)$. In one phase, almost every $G\in(n,m)$ is co-$r$-partite, and in the other phase, almost every $G\in(n,m)$ is the disjoint union of a co-$r$-partite graph and a sparse graph.

— The typical structure of dense claw-free graphs  (2501.17816 - Perkins et al., 29 Jan 2025) in Section 1, subsection “Future Work,” Conjecture labeled Conjecture \ref{conj:Kr1rplus1}