Sub-quorum colorings of graphs

Abstract

A sub-quorum coloring is a partial vertex coloring in which every colored vertex sees at least half of its colored closed neighborhood in its own color. Hedetniemi, Hedetniemi, Laskar and Mulder introduced its maximum number of colors, $\psq(G)$, as an open direction in their foundational work on quorum colorings. We establish general bounds, relate $\psq$ to $2$-independence, discuss computational complexity, and determine exact values for several classical families. For rectangular grids $G_{m,n}=P_m\square P_n$, we give a new profile proof of the known dissociation-number formula, equivalent to earlier exact $3$-path vertex-cover results. The proof supplies equality and rigidity information used to establish the same formula for the auxiliary parameter when the representative matching is restricted to one direction. We also obtain a five-sixths inequality for mixed-direction matchings on even-by-even rectangles. Exact transfer certificates establish the sub-quorum coloring formula for all fixed strip widths $2\le m\le11$. For hypercubes, we prove the dimension-free identity $\psq(Q_n)=\bii(Q_n)=2^{n-1}$ for every $n\ge2$. The upper bound for the sub-quorum coloring number follows from Huang's signed adjacency matrix through a restricted energy estimate and an injective linear map. The computer-assisted grid claims use integer arithmetic and are independently reproducible by the accompanying verifier.