78 research outputs found

    The Radio Number of Grid Graphs

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    The radio number problem uses a graph-theoretical model to simulate optimal frequency assignments on wireless networks. A radio labeling of a connected graph GG is a function f:V(G)Z0+f:V(G) \to \mathbb Z_{0}^+ such that for every pair of vertices u,vV(G)u,v \in V(G), we have f(u)f(v)diam(G)+1d(u,v)\lvert f(u)-f(v)\rvert \ge \text{diam}(G) + 1 - d(u,v) where diam(G)\text{diam}(G) denotes the diameter of GG and d(u,v)d(u,v) the distance between vertices uu and vv. Let span(f)\text{span}(f) be the difference between the greatest label and least label assigned to V(G)V(G). Then, the \textit{radio number} of a graph rn(G)\text{rn}(G) is defined as the minimum value of span(f)\text{span}(f) over all radio labelings of GG. 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 nn grid). In this paper, we completely determine the radio number of the grid graph Ga,bG_{a,b} for a,b>2a,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 labelings of digraphs

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

    Radio Number of Hamming Graphs of Diameter 3

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

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    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

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

    A Lower Bound for Radio kk-chromatic Number of an Arbitrary Graph

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    Radio kk-coloring is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph GG, subject to certain constraints involving the distance between the vertices. Specifically, for any simple connected graph GG with diameter dd and apositive integer kk, 1kd1\leq k \leq d, a radio kk-coloring of GG is an assignment ff of positive integers to the vertices of GG such that f(u)f(v)1+kd(u,v)|f(u)-f(v)|\geq 1+k-d(u, v), where uu and vv are any two distinct vertices of GG and d(u,v)d(u, v) is the distance between uu and vv.In this paper we give a lower bound for the radio kk-chromatic number of an arbitrarygraph in terms of kk, the total number of vertices nn and apositive integer MM such that d(u,v)+d(v,w)+d(u,w)Md(u,v)+d(v,w)+d(u,w)\leq M for all u,v,wV(G)u,v,w\in V(G). If MM is the triameter we get a better lower bound. We also find the triameter MM for several graphs, and show that the lower bound obtained for these graphs is sharp for the case k=dk=d

    Improved bounds on the L(2,1)-number of direct and strong products of graphs

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    2007-2008 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

    On Regular Graphs Optimally Labeled with a Condition at Distance Two

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    For positive integers jkj \geq k, the λj,k\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 λj,k\lambda_{j,k}-number of any r-regular graph is no less than the λj,k\lambda_{j,k}-number of the infinite r-regular tree T(r)T_{\infty}(r). Defining an r-regular graph G to be (j,k,r)(j,k,r)-optimal if and only if λj,k(G)=λj,k(T(r))\lambda_{j,k}(G) = \lambda_{j,k}(T_{\infty}(r)), we establish the equivalence between (j,k,r)(j,k,r)-optimal graphs and r-regular bipartite graphs with a certain edge coloring property for the case jk3˘er{j \over k} \u3e r. The structure of rr-regular optimal graphs for jkr{j \over k} \leq r is investigated, with special attention to jk=1,2{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 λj,k\lambda_{j,k}-numbers of prisms

    L(2,1)-labelings of Cartesian products of two cycles

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    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
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