arXiv subjects

Rafik Sahbi

Publications and source records attributed to Rafik Sahbi.

2 recordsLinked to original sources

arXiv2026

Sub-quorum colorings of graphs

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. The notion was proposed by Hedetniemi, Hedetniemi, Laskar and Mulder as an open direction in their foundational work on quorum colorings. We further develop the study of the sub-quorum coloring number $\psq(G)$ introduced by Hedetniemi, Hedetniemi, Laskar and Mulder. Our emphasis is on structural bounds, computational complexity, grid graphs and hypercubes. We establish general bounds, relate $\psq$ to $2$-independence, discuss computational complexity, determine exact values for several classical families, and give exact and computer-assisted results for grid strips. We also investigate hypercubes. In particular, exact certificates for $Q_n$, $2\le n\le6$, give $\psq(Q_n)=\bii(Q_n)=2^{n-1}$, and we formulate the general equality as a conjecture. The computational claims use integer arithmetic and are independently reproducible by the verifier accompanying the manuscript.

math.CO↗

arXiv2026

Quorum colorings of maximum cardinality in linear time for a subclass of perfect trees

A partition $π=\{V_{1},V_{2},...,V_{k}\}$ of the vertex set $V$ of a graph $G$ into $k$ color classes $V_{i}$, with $1\leq i\leq k$ is called a quorum coloring of $G$ if for every vertex $v\in V$, at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v$. The maximum cardinality of a quorum coloring of $G$ is called the quorum coloring number of $G$ and is denoted by $ψ_{q}(G)$. A quorum coloring of order $ψ_{q}(G)$ is a $ψ_{q}$-coloring. The determination of the quorum coloring number or design a linear-time algorithm computing it in a perfect $N$-ary tree has been posed recently as an open problem by Sahbi. In this paper, we answer this problem by designing a linear-time algorithm for finding both a $ψ_{q}$-coloring and the quorum coloring number for every perfect tree whose the vertices at the same depth have the same degree.

math.CO↗