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KRAL, Daniel; SKREKOVSKI, Riste
SIAM journal on discrete mathematics, 01/2003, Volume: 16, Issue: 3Journal Article
A list channel assignment problem is a triple (G,L,w), where G is a graph, L is a function which assigns to each vertex of G a list of integers (colors), and w is a function which assigns to each edge of G a positive integer (its weight). A coloring c of the vertices of G is proper if c(v)\in L(v)$ for each vertex v and $|c(u)-c(v)|\ge w(uv)$ for each edge uv. A weighted degree $\deg_w(v)$ of a vertex v is the sum of the weights of the edges incident with v. If G is connected, $|L(v)|>\deg_w(v)$ for at least one v, and $|L(v)|\ge\deg_w(v)$ for all v, then a proper coloring always exists. A list channel assignment problem is balanced if $|L(v)|=\deg_w(v)$ for all v. We characterize all balanced list channel assignment problems (G,L,w) which admit a proper coloring. An application of this result is that each graph with maximum degree $\Delta\ge 2$ has an L(2,1)-labeling using integers $0,\ldots,\Delta^2+\Delta-1$.
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