Results 1 
4 of
4
ConflictFree Coloring and its Applications
, 2010
"... Let H = (V, E) be a hypergraph. A conflictfree coloring of H is an assignment of colors to V such that in each hyperedge e ∈ E there is at least one uniquelycolored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to c ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
Let H = (V, E) be a hypergraph. A conflictfree coloring of H is an assignment of colors to V such that in each hyperedge e ∈ E there is at least one uniquelycolored 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, and has been the focus of many recent research papers. In this paper, we survey this notion and its combinatorial and algorithmic aspects.
CoverDecomposition and Polychromatic Numbers
"... A colouring of a hypergraph’s vertices is polychromatic if every hyperedge contains at leastone vertex ofeach colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the coverdecomposition number, is the maximum number of disjoint hyperedge covers. In geomet ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
A colouring of a hypergraph’s vertices is polychromatic if every hyperedge contains at leastone vertex ofeach colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the coverdecomposition number, is the maximum number of disjoint hyperedge covers. In geometric hypergraphs, there is extensive work on lowerbounding these numbers in terms of their trivial upper bounds (minimum hyperedge size and degree); our goal here is to broaden the study beyond geometric settings. We obtain algorithms yielding neartight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded VCdimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for coverdecomposition. Finally, we discuss the generalization of coverdecomposition to sensor cover.
BROOKS TYPE RESULTS FOR CONFLICTFREE COLORINGS AND {a, b}FACTORS IN GRAPHS.
"... Abstract. A vertexcoloring of a hypergraph is conflictfree, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let f(r,∆) be the smallest integer k such that each runiform hypergraph of maximum vertex degree ∆ has a conflictfree coloring with at most k c ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. A vertexcoloring of a hypergraph is conflictfree, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let f(r,∆) be the smallest integer k such that each runiform hypergraph of maximum vertex degree ∆ has a conflictfree coloring with at most k colors. As shown by Tardos and Pach, similarly to a classical Brooks ’ type theorem for hypergraphs, f(r,∆) ≤ ∆+1. Compared to Brooks’ theorem, according to which there is only a couple of graphs/hypergraphs that attain the ∆+1 bound, we show that there are several infinite classes of uniform hypergraphs for which the upper bound is attained. We give better upper bounds in terms of ∆ for large ∆ and establish the connection between conflictfree colorings and socalled {t, r−t}factors in rregular graphs. Here, a {t, r − t}factor is a factor in which each degree is either t or r − t. Among others, we disprove a conjecture of Akbari and Kano [1] stating that there is a {t, r − t}factor in every rregular graph for odd r and any odd t < r 3