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. We study the maximum number of colors $\psq(G)$ from structural, algorithmic and extremal viewpoints. Every $2$-independent set yields a sub-quorum coloring, so $\psq(G)\ge\beta_2(G)$. We prove the degree bound $\psq(G)\le \Delta(G)|V(G)|/(\Delta(G)+\delta(G)-1)$ for $\delta(G)\ge2$, and obtain an exact matching characterization for graphs of maximum degree at most three. Consequently every cubic graph of order $n$ satisfies $n/2\le\psq(G)\le3n/5$, with a stability identity at the upper endpoint. For trees we give an edge representation and a linear-time dynamic program that computes both $\psq(T)$ and an optimal coloring; we also show on the double star $S_{2,2}$ that $\psq$ can exceed the dissociation number even on trees. We prove NP-completeness even for bipartite graphs and determine exact values for several classical families. For rectangular grids, we formulate the conjecture $\psq(G_{m,n})=\beta_2(G_{m,n})=F(m,n)$, recover the dissociation-number formula from known $3$-path vertex-cover results, prove the ladder case directly, and verify the conjecture for every fixed width $2\le m\le11$; widths $3$ through $11$ use exact transfer certificates. For hypercubes we prove $\psq(Q_n)=\beta_2(Q_n)=2^{n-1}$ for every $n\ge2$. Finally, a Nordhaus--Gaddum argument gives $\psq(G)+\psq(\overline G)>\log_2 |V(G)|$ and leads to a complementary upper-bound problem.