Multicoloring unit disk graphs on triangular lattice points

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