Labeling graphs with a condition at distance two
نویسنده
چکیده
The problem arises from the channel assignment problem. An L(2, 1)-labeling of a graph G = (V,E) is a function f : V → {0, 1, 2, . . .} such that |f(x) − f(y)| ≤ 2 if d(x, y) = 2 and |f(x) − f(y)| ≤ 1 if d(x, y) = 2. The span of an L(2, 1)-labeling is the maximum value f(x) among all vertices x in V . The L(2, 1)-labeling number of G, denoted by λ(G) is the minimum span of an L(2, 1)-labeling. A more general setting is as follows. For positive integers j1 ≥ j2 ≥ . . . ≥ jr, an L(j1, j2, . . . , jr)-labeling of a graph G = (V,E) is a function f : V → {0, 1, 2, . . .} such that |f(x) − f(y)| ≤ ji if d(x, y) = i for 1 ≤ i ≤ r. The span of an L(j1, j2, . . . , jr)labeling is the maximum value f(x) among all vertices x in V . The L(j1, j2, . . . , jr)labeling number ofG, denoted by λj1,j2,...,jr(G) is the minimum span of an L(j1, j2, . . . , jr)labeling. Notice that λ(G) = λ2,1(G). Most frequently studied case is when r = 2.
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