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Grad and classes with bounded expansion I. Decompositions
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European Journal of Combinatorics
Grad and classes with bounded expansion III. Restricted graph homomorphism dualities
European Journal of Combinatorics
On forbidden subdivision characterizations of graph classes
European Journal of Combinatorics
Deciding First-Order Properties for Sparse Graphs
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
European Journal of Combinatorics
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Consider a graph G=(V,E) and, for each vertex v@?V, a subset @S(v) of neighbors of v. A @S-coloring is a coloring of the elements of V so that vertices appearing together in some @S(v) receive pairwise distinct colors. An obvious lower bound for the minimum number of colors in such a coloring is the maximum size of a set @S(v), denoted by @r(@S). In this paper we study graph classes F for which there is a function f, such that for any graph G@?F and any @S, there is a @S-coloring using at most f(@r(@S)) colors. It is proved that if such a function exists for a class F, then f can be taken to be a linear function. It is also shown that such classes are precisely the classes having bounded star chromatic number. We also investigate the list version and the clique version of this problem, and relate the existence of functions bounding those parameters to the recently introduced concepts of classes of bounded expansion and nowhere-dense classes.