The $L(2,1)$-Labeling Problem on Graphs

Abstract

An L(2, 1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that If(x)f(Y)l-> 2 if d(x, y) 1 and If(x) f(Y)l-> i if d(x,y) 2. The 5(2, 1)-labeling number /k(G) of G is the smallest number k such that G has an 5(2, 1)-labeling with max{f(v):v E V(G)} k. In this paper, we give exact formulas of A(G 2 H) and A(G + H). We also prove that A(G) _ A 2 + A for any graph G of maximum degree A. For odd-sun-free (OSF)-chordal graphs, the upper bound can be reduced to A(G) _ 2A + 1. For sun-free (SF)-chordal graphs, the upper bound can be reduced to A(G) _ A + 2X(G 2. Finally, we present a polynomial time algorithm to determine A(T) for a tree T. Key words. L(2, 1)-labeling, T-coloring, union, join, chordal graph, perfect graph, tree, bipartite matching, algorithm AMS subject classifications. 05C15, 05C78 1. Introduction. The channel assignment problem is to assign a channel (nonnegative integer) to each radio transmitter so that interfering transmitters are assigned channels whose separation is not in a set of disallowed separations. Hale [11] formulated this problem into the notion of the T-coloring of a graph, and the T-coloring problem has been extensively studied over the past decade (see [4, 5, 7, 13, 14, 16, 17, 19]). Roberts [15] proposed a variation of the channel assignment problem in which "close" transmitters must receive different channels and "very close" transmitters must receive channels that are at least two channels apart. To formulate the problem in graphs, the transmitters are represented by the vertices of a graph; two vertices are "very close" if they are adjacent in the graph and "close" if they are of distance two in the graph. More precisely, an L(2, 1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that If(x)-f(Y)l >-2 if d(x, y) 1 and If(x) f(Y)l-> 1 if d(x, y) 2. A k-L(2, 1)-labeling is an L(2, 1)labeling such that no label is greater than k. The L(2, 1)-labeling number of G, denoted by A(G), is the smallest number k such that G has a k-L(2, 1)-labeling. Griggs and Yeh [10] and Yeh [21] determined the exact values of A(P), A(C), and A(W), where P is a path of n vertices, Cn is a cycle of n vertices, and Wn is an n-wheel obtained from Cn by adding a new vertex adjacent to all vertices in C. For the n-cube Q, Jonas [12] showed that n + 3 _< A(Q). Griggs and Yeh [10] showed that i(Qn) _< 2n + 1 for n _> 5. They also determined A(Q) for n _< 5 and conjectured that the lower bound n + 3 is the actual value of A(Q) for n _> 3. Using a coding theory method, Whittlesey, Georges, and Mauro [20] proved that /(Qn) _ 2 k + 2 k-q+1 2, where n _< 2 k-q and 1 _< q _< k + 1. In particular, A(Q2k-k-1) _< 2 } 1. As a consequence, A(Qn) _< 2n for n _> 3.

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