Attainment of the minimum-degree-five repetition bound

Determine whether the lower bound \(\mathrm{MP}(n,5,0)\geq (n+12)/3\) is attained for every order \(n\equiv 0\pmod 3\) with \(n>48\).

Background

For maximal planar graphs of minimum degree five, the paper proves the lower bound MP(n,5,0)≥⌈(n+12)/3⌉\mathrm{MP}(n,5,0)\geq \lceil(n+12)/3\rceil. Equality requires the highly constrained degree sequence 5s6s7s−125^s6^s7^{s-12}, where s=(n+12)/3s=(n+12)/3.

The authors construct maximal planar graphs attaining this bound for every multiple of three between 24 and 48, but do not establish whether the same equality continues for all larger multiples of three. This is a concrete realization problem for prescribed degree sequences in maximal planar graphs.

References

Whether it is attained for every n\equiv 0\ (\operatorname{mod}\ 3) with n>48 remains open.

— Spreads of degrees in maximal planar graphs -- an exposition  (2610.11395 - Caro et al., 8 Oct 2026) in Section 2.1, immediately following Theorem 2.3 (Theorem \ref{Tm_hex})

For which pairs $(\delta ,k)$ is $\mathrm{MP}(n,\delta ,k)-c_{\delta ,k}n$ bounded, and is it eventually periodic in $n$?

— Spreads of degrees in maximal planar graphs -- an exposition  (2610.11395 - Caro et al., 8 Oct 2026) in Section 7, Problem \ref{Pr_error}

Is $\mathrm{MP}(n,3,k)\leq \mathrm{MP}(n,4,k)\leq \mathrm{MP}(n,5,k)$ for every $k\geq 0$ and every $n$ at which all three are defined?

— Spreads of degrees in maximal planar graphs -- an exposition  (2610.11395 - Caro et al., 8 Oct 2026) in Section 7, Problem \ref{Pr_monotone}