82 research outputs found

    Distance-two labelings of digraphs

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    For positive integers j≥kj\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 λ⃗j,k\vec{\lambda}_{j,k}-number λ⃗j,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 λ⃗j,k\vec{\lambda}_{j,k}- numbers of digraphs. In particular, we determine λ⃗j,k\vec{\lambda}_{j,k}- numbers of digraphs whose longest dipath is of length at most 2, and λ⃗j,k\vec{\lambda}_{j,k}-numbers of ditrees having dipaths of length 4. We also give bounds for λ⃗j,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 λ⃗j,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

    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,v∈V(G)u,v \in V(G), we have ∣f(u)−f(v)∣≥diam(G)+1−d(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

    On n-fold L(j,k)-and circular L(j,k)-labelings of graphs

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    AbstractWe initiate research on the multiple distance 2 labeling of graphs in this paper.Let n,j,k be positive integers. An n-fold L(j,k)-labeling of a graph G is an assignment f of sets of nonnegative integers of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), |a−b|≥j if uv∈E(G), and |a−b|≥k if u and v are distance 2 apart. The span of f is the absolute difference between the maximum and minimum integers used by f. The n-fold L(j,k)-labeling number of G is the minimum span over all n-fold L(j,k)-labelings of G.Let n,j,k and m be positive integers. An n-fold circular m-L(j,k)-labeling of a graph G is an assignment f of subsets of {0,1,…,m−1} of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), min{|a−b|,m−|a−b|}≥j if uv∈E(G), and min{|a−b|,m−|a−b|}≥k if u and v are distance 2 apart. The minimum m such that G has an n-fold circular m-L(j,k)-labeling is called the n-fold circular L(j,k)-labeling number of G.We investigate the basic properties of n-fold L(j,k)-labelings and circular L(j,k)-labelings of graphs. The n-fold circular L(j,k)-labeling numbers of trees, and the hexagonal and p-dimensional square lattices are determined. The upper and lower bounds for the n-fold L(j,k)-labeling numbers of trees are obtained. In most cases, these bounds are attainable. In particular, when k=1 both the lower and the upper bounds are sharp. In many cases, the n-fold L(j,k)-labeling numbers of the hexagonal and p-dimensional square lattices are determined. In other cases, upper and lower bounds are provided. In particular, we obtain the exact values of the n-fold L(j,1)-labeling numbers of the hexagonal and p-dimensional square lattices

    Radio Labelings of Distance Graphs

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    A radio kk-labeling of a connected graph GG is an assignment cc of non negative integers to the vertices of GG such that ∣c(x)−c(y)∣≥k+1−d(x,y),|c(x) - c(y)| \geq k+1 - d(x,y), for any two vertices xx and yy, x≠yx\ne y, where d(x,y)d(x,y) is the distance between xx and yy in GG. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set Z\Z of integers as vertex set and in which two distinct vertices i,j∈Zi, j \in \Z are adjacent if and only if ∣i−j∣∈D|i - j| \in D.Comment: 14 page

    On Regular Graphs Optimally Labeled with a Condition at Distance Two

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    For positive integers j≥kj \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 jk≤r{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

    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, 1≤k≤d1\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+k−d(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,w∈V(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
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