Conflict-Free Coloring and its Applications [chapter]

Shakhar Smorodinsky
2013 Bolyai Society Mathematical Studies  
Let H = (V, E) be a hypergraph. A conflict-free coloring of H is an assignment of colors to V such that, in each hyperedge e ∈ E, there is at least one uniquely-colored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to cellular antennae, in battery consumption aspects of sensor networks, in RFID protocols, and several other fields. Conflict-free coloring has been the focus of many recent research papers. In this
more » ... per, we survey this notion and its combinatorial and algorithmic aspects. large gap between χ cf (H) and χ um (H). Consider, for example, two sets A and B each of cardinality n > 1. Let H = (A ∪ B, E) where E consists of all triples of elements e such that e ∩ A = ∅ and e ∩ B = ∅. In other words E consists of all triples containing two elements from one of the sets A or B and one element from the other set. It is easily seen that χ cf (H) = 2 by simply coloring all elements of A with 1 and all elements of B with 2. It is also not hard to verify that χ um (H) ≥ n (in fact χ um (H) = n + 1). Indeed, let C be a UM-coloring of H. If all elements of A are colored with distinct colors we are done. Otherwise, there exist two elements u, v in A with the same color, say i. We claim that all elements of B are colored with colors greater than i. Assume to the contrary that there is an element w ∈ B with color C(w) = j ≤ i. However, in that case the hyperedge {u, v, w} does not have the unique-maximum property. Hence all colors of B are distinct for otherwise if there are two vertices w 1 , w 2 with the same color, again the hyperedge {w 1 , w 2 , u} does not have the unique-maximum property. Let us describe a simple yet an important example of a hypergraph H and analyze its chromatic number χ(H) and its CF-chromatic number χ cf (H). The vertices of the hypergraph consist of the first n integers [n] = {1, . . . , n}. The hyperedge-set is the set of all (non-empty) subsets of [n] consisting of consecutive elements of [n], e.g., {2, 3, 4}, {2}, the set [n], etc. We refer to such hypergraphs as hypergraphs induced by points on the line with respect to intervals or as the discrete intervals hypergraph. Trivially, we have χ(H) = 2. We will prove the following proposition: Proposition 1.2. χ cf (H) = χ um (H) = log n + 1.
doi:10.1007/978-3-642-41498-5_12 fatcat:gjf6tvt5yvgn3hlhuc3fcj3rda