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Distance Three Labelings of Trees∗
"... An L(2, 1, 1)labeling of a graph G assigns nonnegative integers to the vertices of G in such a way that labels of adjacent vertices differ by at least two, while vertices that are at distance at most three are assigned different labels. The maximum label used is called the span of the labeling, and ..."
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An L(2, 1, 1)labeling of a graph G assigns nonnegative integers to the vertices of G in such a way that labels of adjacent vertices differ by at least two, while vertices that are at distance at most three are assigned different labels. The maximum label used is called the span of the labeling, and the aim is to minimize this value. We show that the minimum span of an L(2, 1, 1)labeling of a tree can be bounded by a lower and an upper bound with difference one. Moreover, we show that deciding whether the minimum span attains the lower bound is an NPcomplete problem. This answers a known open problem, which was recently posed by King, Ras, and Zhou as well. We extend some of our results to general graphs and/or to more general distance constraints on the labeling. 1
Labeling DotCartesian and DotLexicographic Product Graphs with a Condition at Distance Two
"... If d(x, y) denotes the distance between vertices x and y in a graph G, then an L(2, 1)labeling of a graph G is a function f from vertices of G to nonnegative integers such that f(x)−f(y)  ≥ 2 if d(x, y) = 1, and f(x)−f(y)  ≥ 1 if d(x, y) = 2. Griggs and Yeh conjectured that for any graph wi ..."
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If d(x, y) denotes the distance between vertices x and y in a graph G, then an L(2, 1)labeling of a graph G is a function f from vertices of G to nonnegative integers such that f(x)−f(y)  ≥ 2 if d(x, y) = 1, and f(x)−f(y)  ≥ 1 if d(x, y) = 2. Griggs and Yeh conjectured that for any graph with maximum degree ∆ ≥ 2, there is an L(2, 1)labeling with all labels not greater than ∆². We prove that the conjecture holds for dotCartesian products and dotlexicographic products of two graphs with possible minor exceptions in some special cases. The bounds obtained are in general much better than the ∆2bound.