Four-particle monochromatic quantum graphs for all D≥4

For which values of $n$ and $d$ do there exist bi-colored weighted graphs on $n$ vertices and with $d$ different colors such that all $d$ monochromatic inherited vertex colorings have unit weight, and every other inherited vertex coloring has weight zero (cancels out)? Here, for an inherited vertex coloring $c$ , its weight is defined by $w(c):=\sum_{PM\in\mathcal{M}}\prod_{e\in PM}w_e$ , where $\mathcal{M}$ is the set of perfect matchings whose inherited vertex coloring is $c$ .

The problem asks whether complex edge weights can make every monochromatic inherited colouring have unit amplitude and every nonmonochromatic colouring have zero amplitude for four particles and $D\geq4$ . The framework was posed in connection with quantum physics; the cited discussion records no resolution of this complex-weighted case.

Known results

Alternately coloured cycles $C_{2n}$ give affirmative examples for $d=2$ and arbitrary even $n$ .
A decomposition of $K_4$ into three perfect matchings gives the case $d=3$ , $n=4$ .
Ilya Bogdanov ruled out other examples when all edge weights are positive real numbers.
The 2022 paper characterizes a relevant subclass, but does not settle arbitrary complex weights for four particles and $D\geq4$ .

Current status (as of May 2026): the complex-weighted problem for four particles and $D\geq4$ remains open, with no recorded proof or counterexample.

Four-particle monochromatic quantum graphs for all $D\geq4$

Progress summary

Solutions 0