Radiocoloring Graphs via the Probabilistic Method
نویسندگان
چکیده
We employ here the Probabilistic Method, a way of reasoning which shows existence of combinatorial structures and properties to prove refute conjectures. The radiocoloring problem (RCP) is the problem of assigning frequencies to the transmitters of a network so that transmitters of distance one get frequencies that differ by at least two and any two transmitters of distance one get frequencies that differ by at least one. The objective of an assignment may be to minimize the number of frequencies used (order) or the range of them (span). Here, we study the optimization version of RCP where the objective is to minimize the order. In graph theory terms the problem is modelled by a variation of the vertex graph coloring problem. We investigate upper bounds for the minimum number of colors needed in a radiocoloring assignment of a graph G. We first provide an upper bound for the minimum number of colors needed to radiocolor a graph G of girth at most 7. Then, we study whether the minimum order of a radiocoloring assignment is determined by local conditions, i.e. by the minimum order radiocoloring assignment of some small subgraphs of it. We state a related conjecture which is analogous to a theorem of Molloy and Reed for the vertex coloring problem [12]. We then investigate whether the conjecture contradicts a Theorem of Molloy and Reed for the vertex coloring when applied on the graph G2 obtained by G. The graph G2 has the same vertex set as G and there is an edge in G2 between any two vertices iff their distance in G is at most 2. We prove that there is no case where the conjecture stated here contradicts the Theorem of Molloy and Reed. Areas: Graph Theory, Probabilistic Method, Coloring, Radiocoloring, Upper Bounds
منابع مشابه
Radiocoloring in planar graphs: Complexity and approximations
The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters, by exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graphG(V,E) is an assignment function : V → N such that | (u)− (v)| 2, when u, v are neighbors in ...
متن کاملRadiocolorings in Periodic Planar Graphs: PSPACE-Completeness and Efficient Approximations for the Optimal Range of Frequencies
The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V,E) is an assignment function Λ : V → IN such that |Λ(u) − Λ(v)| ≥ 2, when u, v are neighbors...
متن کاملImplementation and Analysis of a Parallel Algorithm for Radiocoloring
A radiocoloring of a graph G is an assignment to each node, one of the colors 0, 1, 2, . . . λ such that all pairs of adjacent nodes, get colors which differ by at least two and no pair of nodes at distance two, get the same color. The radiocoloring problem (RCP) consists of determining the minimum λ for a given graph. In this paper, we first implement the PARC (Parallel Algorithm for Radiocolo...
متن کاملOPTIMAL ANALYSIS OF NON-REGULAR GRAPHS USING THE RESULTS OF REGULAR MODELS VIA AN ITERATIVE METHOD
In this paper an efficient method is developed for the analysis of non-regular graphs which contain regular submodels. A model is called regular if it can be expressed as the product of two or three subgraphs. Efficient decomposition methods are available in the literature for the analysis of some classes of regular models. In the present method, for a non-regular model, first the nodes of th...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2003