Exact wild number for minimum proper colorings of complete graphs

Establish that for every integer n≥3 and every minimum proper edge coloring ρ of the complete graph K_n, the wild number satisfies wld_ρ(K_n)=⌊4⌈n/2⌉/3⌋.

Background

The paper proves the lower bound wld_ρ(K_n)≥⌊4⌈n/2⌉/3⌋ for every minimum proper coloring of K_n. It also establishes equality for all orders 3≤n≤12 by constructing suitable color-connecting sets and proves additional upper bounds for larger odd and even orders. These results motivate the conjecture that the lower bound is sharp universally, independently of the choice of minimum proper coloring.

References

These results lead to the following conjecture. For every $n\ge3$ and every minimum proper coloring $\rho$ of $K_n$, $$ \wld_\rho(K_n) = \left\lfloor \frac{4\lceil n/2\rceil}{3}\right\rfloor. $$

— Wild number for spanning trees in complete and complete bipartite graphs  (2610.22142 - Dawkins et al., 15 Oct 2026) in Conjecture, Section 7, Concluding Remarks (labelled conj:proper-kn-exact)