78 research outputs found
The Radio Number of Grid Graphs
The radio number problem uses a graph-theoretical model to simulate optimal
frequency assignments on wireless networks. A radio labeling of a connected
graph is a function such that for every pair
of vertices , we have where denotes the diameter of and
the distance between vertices and . Let be the
difference between the greatest label and least label assigned to . Then,
the \textit{radio number} of a graph is defined as the minimum
value of over all radio labelings of . So far, there have
been few results on the radio number of the grid graph: In 2009 Calles and
Gomez gave an upper and lower bound for square grids, and in 2008 Flores and
Lewis were unable to completely determine the radio number of the ladder graph
(a 2 by grid). In this paper, we completely determine the radio number of
the grid graph for , characterizing three subcases of the
problem and providing a closed-form solution to each. These results have
implications in the optimization of radio frequency assignment in wireless
networks such as cell towers and environmental sensors.Comment: 17 pages, 7 figure
Distance-two labelings of digraphs
For positive integers , an -labeling of a digraph is a
function from into the set of nonnegative integers such that
if is adjacent to in and if
is of distant two to in . Elements of the image of are called
labels. The -labeling problem is to determine the
-number of a digraph , which
is the minimum of the maximum label used in an -labeling of . This
paper studies - numbers of digraphs. In particular, we
determine - numbers of digraphs whose longest dipath is of
length at most 2, and -numbers of ditrees having dipaths
of length 4. We also give bounds for -numbers of bipartite
digraphs whose longest dipath is of length 3. Finally, we present a linear-time
algorithm for determining -numbers of ditrees whose
longest dipath is of length 3.Comment: 12 pages; presented in SIAM Coference on Discrete Mathematics, June
13-16, 2004, Loews Vanderbilt Plaza Hotel, Nashville, TN, US
Radio Number of Hamming Graphs of Diameter 3
For a simple, connected graph, a vertex labeling is called a \emph{radio labeling of } if it satisfies |f(u)-f(v)|\geq\diam(G)+1-d(u,v) for all distinct vertices . The \emph{radio number of } is the minimal span over all radio labelings of . If a bijective radio labeling onto exists, is called a \emph{radio graceful} graph. We determine the radio number of all diameter 3 Hamming graphs and show that an infinite subset of them is radio graceful
The Research on the L(2,1)-labeling problem from Graph theoretic and Graph Algorithmic Approaches
The L(2,1) -labeling problem has been extensively researched on many graph classes. In this thesis, we have also studied the problem on some particular classes of graphs.
In Chapter 2 we present a new general approach to derive upper bounds for L(2,1)-labeling numbers and applied that approach to derive bounds for the four standard graph products.
In Chapter 3 we study the L(2,1)-labeling number of the composition of n graphs.
In Chapter 4 we consider the Cartesian sum of graphs and derive, both, lower and upper bounds for their L(2,1)-labeling number. We use two different approaches to derive the upper bounds and both approaches improve previously known bounds. We also present new approximation algorithms for the L(2,1 )-labeling problem on Cartesian sum graphs.
In Chapter 5, we characterize d-disk graphs for d\u3e1, and give the first upper bounds on the L(2,1)-labeling number for this class of graphs.
In Chapter 6, we compute upper bounds for the L(2,1)-labeling number of total graphs of K_{1,n}-free graphs.
In Chapter 7, we study the four standard products of graphs using the adjacency matrix analysis approach.
In Chapter 8, we determine the exact value for the L(2,1)-labeling number of a particular class of Mycielski graphs. We also provide, both, lower and upper bounds for the L(2,1)-labeling number of any Mycielski graph
Optimal L(h, k) labelings of Cartesian products of complete graphs and paths
In an labeling of a graph we assign non-negative integers to the vertices of the graph such that the labels of the vertices that are at a distance of one have a difference of at least and the labels of the vertices which are at a distance of two have a difference of at least . The aim in general is to minimize the span, where the span is the difference between highest and lowest label used. In this thesis we analyze labelings of Cartesian products of complete graphs and path. For we establish the minimum span of these graphs. For $
A Lower Bound for Radio -chromatic Number of an Arbitrary Graph
Radio -coloring is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph , subject to certain constraints involving the distance between the vertices. Specifically, for any simple connected graph with diameter and apositive integer , , a radio -coloring of is an assignment of positive integers to the vertices of such that , where and are any two distinct vertices of and is the distance between and .In this paper we give a lower bound for the radio -chromatic number of an arbitrarygraph in terms of , the total number of vertices and apositive integer such that for all . If is the triameter we get a better lower bound. We also find the triameter for several graphs, and show that the lower bound obtained for these graphs is sharp for the case
Improved bounds on the L(2,1)-number of direct and strong products of graphs
2007-2008 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe
On Regular Graphs Optimally Labeled with a Condition at Distance Two
For positive integers , the -number of graph Gis the smallest span among all integer labelings of V(G) such that vertices at distance two receive labels which differ by at least k and adjacent vertices receive labels which differ by at least j. We prove that the -number of any r-regular graph is no less than the -number of the infinite r-regular tree . Defining an r-regular graph G to be -optimal if and only if , we establish the equivalence between -optimal graphs and r-regular bipartite graphs with a certain edge coloring property for the case . The structure of -regular optimal graphs for is investigated, with special attention to . For the latter, we establish that a (2,1,r)-optimal graph, through a series of edge transformations, has a canonical form. Finally, we apply our results on optimality to the derivation of the -numbers of prisms
L(2,1)-labelings of Cartesian products of two cycles
AbstractAn L(2,1)-labeling of a graph is an assignment of nonnegative integers to its vertices so that adjacent vertices get labels at least two apart and vertices at distance two get distinct labels. The λ-number of a graph G, denoted by λ(G), is the minimum range of labels taken over all of its L(2,1)-labelings. We show that the λ-number of the Cartesian product of any two cycles is 6, 7 or 8. In addition, we provide complete characterizations for the products of two cycles with λ-number exactly equal to each one of these values
- …