Pair-Defensive Silver Colourings of Hypercubes

Source abstract

A silver colouring is a proper colouring in which every colour appears in the closed neighbourhood of each vertex of a prescribed independent set. This local condition suffices when vertices are tested one at a time. We study a stronger requirement for simultaneous testing: whenever one or two vertices of the independent set are attacked together, each colour must supply distinct nearby defenders for them. We call this a pair-defensive silver colouring. For the hypercube $Q_d$, with one parity class as the attacked set, we prove that the maximum number of colours in a pair-defensive silver colouring is at most $\lfloor(d+3)/2\rfloor$, roughly half the ordinary silver-colouring target $d+1$. We construct colourings attaining this bound in four consecutive dimensions around every power of two, and we study the structure of the extremal, bound-attaining colourings. In each odd critical dimension, we characterize the extremal colourings by a partition of the defender parity into regular, triangle-free subgraphs of the halved cube. This characterization also has a local form in terms of coordinate matchings and a defect coordinate