Determine the maximum number of factors for connected graphs

Determine the maximum number of $F$-factors, up to a subexponential multiplicative factor, in an $n$-vertex, $m$-edge graph for every connected graph $F$.

Background

The paper establishes asymptotically sharp Kahn–Lovász-type degree-sequence inequalities and Kruskal–Katona-type bounds for FF-factors when FF is Hamiltonian, as well as for certain connected non-Hamiltonian graphs containing two vertex-disjoint cycles of equal length spanning their vertex set. For general connected graphs, however, Theorem 3.1 supplies an upper bound while the constructions in Section 2 can yield substantially different lower bounds, leaving a considerable gap in both the degree-sequence and prescribed-edge settings. The stated problem asks for the correct asymptotic order of NF(n,m)N_F(n,m) for every connected graph FF, allowing a subexponential multiplicative discrepancy.

References

Although Theorem~\ref{thm:general-factors} provides a Kahn--Lov{a}sz-type inequality for every connected graph $F$, there remains a considerable gap between the upper and lower bounds in general, both for the degree-sequence setting and for the corresponding Kruskal--Katona-type problem. This naturally leads to the following problem.

\begin{problem}\label{prob:kruskal-katona-type} Determine $N_F(n,m)$ up to a subexponential multiplicative factor for every connected graph $F$. \end{problem}

— Kahn--Lovász-type inequalities for graph factors  (2608.20303 - Lee, 20 Aug 2026) in Problem 1, Section 5 (Concluding Remarks)

We believe that the same phenomenon should hold more generally for every connected graph $F$ admitting a perfect fractional matching. We conclude the paper with the following conjecture.

\begin{conjecture}\label{conj:KL-pfm} For every $ > 0$ and a connected graph $F$ admitting a perfect fractional matching, there exists a constant $C = C(F, ) > 0$ such that the following holds. For all $n$-vertex graph $G$, we have \begin{equation*} N_{\mathrm{factor}(F; G) \leq \left( (1 + ) \frac{|V(F)|}{|\mathrm{Aut}(F)|} \right){\frac{n}{|V(F)|} \cdot \left( \prod_{v\in V(G)} \frac{d_G(v) + C}{e} \right){1 - \frac{1}{|V(F)|}. \end{equation*} \end{conjecture}

— Kahn--Lovász-type inequalities for graph factors  (2608.20303 - Lee, 20 Aug 2026) in Conjecture 2, Section 5 (Concluding Remarks)