The Combinatorial k-Deck

Abstract

We consider colouring reconstruction problems for a finite group G acting on a finite set X. For a given set F of colours the G-action on X induces a G-action on the set $F^X=\{c:X\longrightarrow F\}$ of colourings of X as well as on the set $\bigcup _{K\subseteq X}F^K$ of partial colourings of X. The combinatorial k-deck of a colouring $c\in F^X$ consists of the multiset of G-classes of the restrictions of c to all k-element subsets of X. The problem is to reconstruct the G-class of the full colouring c from the partial information given by the combinatorial k-deck. This new notion of deck generalizes the well-known k-deck for subsets in finite groups and is a natural alternative to another (more analytical) notion of deck for real- or integer-valued functions considered by various authors. We compare these kinds of decks and show that the combinatorial deck is to arbitrary high degree stronger than the traditional analytical one. However, we also show that even for the combinatorial deck there is no global colouring reconstruction number for all finite groups. Additionally, we demonstrate that a global subset reconstruction number (if it exists) has to be at least 6.