MATHEMATICAL ENGINEERING TECHNICAL REPORTS Multicoloring Unit Disk Graphs on Triangular Lattice Points

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) def. = {(xe1 + ye2) | x ∈ {0, 1, . . . , m − 1}, y ∈ {0, 1, . . . , n − 1}} where e1 def. = (1, 0), e2 def. = (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. Given a non-negative vertex weight vector w ∈ Z (m,n) + , a multicoloring of (Tm,n(d),w) 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 propose a polynomial time approximation algorithm for multicoloring (Tm,n(d),w). Our algorithm is based on the well-solvable cases that the graph Tm,n(d) is a perfect graph. We also showed a necessary and sufficient condition that Tm,n(d) is perfect. For any d ≥ 1, our algorithm finds a multicoloring which uses at most α(d)ω+O(d) colors, where ω denotes the weighted clique number. When d = 1, √ 3, 2, √ 7, 3, 1 Supported by Superrobust Computation Project of the 21st Century COE Program “Information Science and Technology Strategic Core.”