Interpolation profiles and agreement quotas beyond odd cycles

Characterize graph–palette pairs for which every extendable nonempty partial coloring admits every integer profile coordinatewise between two of its extension profiles, subject to the fixed total, and determine sharp simultaneous per-color agreement quotas for graphs of bounded treewidth with a fixed palette when a feasible target profile is prescribed.

Background

The paper proves an interpolation theorem for odd cycles: given an extendable partial coloring and two extension profiles, every integer profile coordinatewise between them, with the same total number of vertices, is also realized by an extension. The authors ask for a characterization of the graph–palette settings in which this phenomenon holds more generally.

The problem also asks for sharp simultaneous per-color agreement guarantees on graphs of bounded treewidth when the target coloring must have prescribed color multiplicities. This extends the paper’s odd-cycle agreement-quota results and differs from flexible list-coloring results because the target profile is fixed in advance.