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 $G$ is a function $f:V(G) \to \mathbb Z_{0}^+$ such that for every pair of vertices $u,v \in V(G)$, we have $\lvert f(u)-f(v)\rvert \ge \text{diam}(G) + 1 - d(u,v)$ where $\text{diam}(G)$ denotes the diameter of $G$ and $d(u,v)$ the distance between vertices $u$ and $v$. Let $\text{span}(f)$ be the difference between the greatest label and least label assigned to $V(G)$. Then, the \textit{radio number} of a graph $\text{rn}(G)$ is defined as the minimum value of $\text{span}(f)$ over all radio labelings of $G$. 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 $n$ grid). In this paper, we completely determine the radio number of the grid graph $G_{a,b}$ for $a,b>2$, 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 labeling of direct product of paths and cycles

Suppose that $[n]=\left\{0,1,2,...,n\right\}$ is a set of non-negative integers and $h,k \in [n]$. The $L(h,k)$-labeling of graph $G$ is the function $l:V(G)\rightarrow[n]$ such that $\left|l(u)-l(v)\right|\geq h$ if the distance $d(u,v)$ between $u$ and $v$ is one and $\left|l(u)-l(v)\right| \geq k$ if the distance $d(u,v)$ is two. Let $L(V(G))=\left\{l(v): v \in V(G)\right\}$ and let $p$ be the maximum value of $L(V(G)).$ Then $p$ is called $\lambda_h^k-$number of $G$ if $p$ is the least possible member of $[n]$ such that $G$ maintains an $L(h,k)-$labeling. In this paper, we establish $\lambda_1^1-$ numbers of $P _m \times C_n$ graphs for all $m \geq 2$ and $n\geq 3$.Comment: 13 pages, 9 figure

Distance-two labelings of digraphs

For positive integers $j\ge k$, an $L(j,k)$-labeling of a digraph $D$ is a function $f$ from $V(D)$ into the set of nonnegative integers such that $|f(x)-f(y)|\ge j$ if $x$ is adjacent to $y$ in $D$ and $|f(x)-f(y)|\ge k$ if $x$ is of distant two to $y$ in $D$. Elements of the image of $f$ are called labels. The $L(j,k)$-labeling problem is to determine the $\vec{\lambda}_{j,k}$-number $\vec{\lambda}_{j,k}(D)$ of a digraph $D$, which is the minimum of the maximum label used in an $L(j,k)$-labeling of $D$. This paper studies $\vec{\lambda}_{j,k}$- numbers of digraphs. In particular, we determine $\vec{\lambda}_{j,k}$- numbers of digraphs whose longest dipath is of length at most 2, and $\vec{\lambda}_{j,k}$-numbers of ditrees having dipaths of length 4. We also give bounds for $\vec{\lambda}_{j,k}$-numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining $\vec{\lambda}_{j,1}$-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

Optimal L(h, k) labelings of Cartesian products of complete graphs and paths

In an $L(h,k)$ labeling of a graph $G$ 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 $h$ and the labels of the vertices which are at a distance of two have a difference of at least $k$. The aim in general is to minimize the $L(h,k)$ span, where the $L(h,k)$ span is the difference between highest and lowest label used. In this thesis we analyze $L(h,k)$ labelings of Cartesian products of complete graphs and path. For $h \geq k$ we establish the minimum $L(h,k)$ span of these graphs. For $

Radio Number of Hamming Graphs of Diameter 3

For $G$ a simple, connected graph, a vertex labeling $f:V(G)\to \Z_+$ is called a \emph{radio labeling of $G$} if it satisfies |f(u)-f(v)|\geq\diam(G)+1-d(u,v) for all distinct vertices $u,v\in V(G)$. The \emph{radio number of $G$} is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto $\{1,2,\dots,|V(G)|\}$ exists, $G$ 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

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 $j \geq k$, the $\lambda_{j,k}$-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 $\lambda_{j,k}$-number of any r-regular graph is no less than the $\lambda_{j,k}$-number of the infinite r-regular tree $T_{\infty}(r)$. Defining an r-regular graph G to be $(j,k,r)$-optimal if and only if $\lambda_{j,k}(G) = \lambda_{j,k}(T_{\infty}(r))$, we establish the equivalence between $(j,k,r)$-optimal graphs and r-regular bipartite graphs with a certain edge coloring property for the case ${j \over k} \u3e r$. The structure of $r$-regular optimal graphs for ${j \over k} \leq r$ is investigated, with special attention to ${j \over k} = 1,2$. 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 $\lambda_{j,k}$-numbers of prisms