Existence of a 5-regular bone-idle graph

Determine whether there exists a 5-regular bone-idle graph, where bone-idleness means that every edge has zero Ollivier–Ricci curvature for every idleness parameter.

Background

Bone-idleness is a stronger property than Lin–Lu–Yau Ricci-flatness: an edge is bone-idle when its zero-idleness Ollivier–Ricci curvature and its Lin–Lu–Yau curvature both vanish. The paper reviews prior results showing that no 3-regular bone-idle graph exists, that 4-regular bone-idle graphs have been completely classified, and that no symmetric 5-regular graph or Cartesian product of a 3-regular graph and a 2-regular graph is bone-idle.

The authors construct the family H_q of connected 5-regular Lin–Lu–Yau Ricci-flat graphs and prove that these graphs are not bone-idle. Thus, the construction does not resolve the broader existence question for 5-regular bone-idle graphs, which remains open.