Multicoloring unit disk graphs on triangular lattice points

摘要

Given a pair of non-negative integers m and n, P(m, n) denotes a subset of 2-dimensional triangular lattice points defined by P(m, n) [EQUATION], {(xe1 +ye2 | x ∈ {0, 1,..., m -1}, y ∈ {0, 1,..., n -1}} where e1 [EQUATION] (1,0), e2 [EQUATION] (1/2, √3/2). Let Tm,n(d) be an undirected graph defined on vertex set P(m, n) satisfying that two vertices are adjacent if and only if the Euclidean distance between the pair is less than or equal to d. This paper discusses a necessary and sufficient condition that Tm,n(d) is perfect; we show that [∀m ∈ Z+ Tm,n(d) is perfect ] if and only if d ≥ √n² -3n + 3.Given a non-negative vertex weight vector w ∈ Z^p(m,n)+ a multicoloring of (Tm,n(d), ω) is an assignment of colors to P(m, n) such that each vertex v ∈ P(m, n) admits w(v) colors and every adjacent pair of two vertices does not share a common color. We also give an efficient algorithm for multicoloring (Tm, n (d), w) when P(m, n) is perfect.In general case, our results on the perfectness of P(m, n) implies a polynomial time approximation algorithm for multicoloring (Tm, n (d), w). Our algorithm finds a multicoloring which uses at most α(dω + O(d³) colors, where ω denotes the weighted clique number. When d = 1, √3, 2, √7, 3, the approximation ratio α(d) = (4/3), (5/3), (7/4), (5/3), (7/4), respectively. When d 1, we showed that α(d) ≤ (1 + 2/√3 +2√3-3/d) 1 + 2/√3 2.155.