7,932 research outputs found
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
Label Placement in Road Maps
A road map can be interpreted as a graph embedded in the plane, in which each
vertex corresponds to a road junction and each edge to a particular road
section. We consider the cartographic problem to place non-overlapping road
labels along the edges so that as many road sections as possible are identified
by their name, i.e., covered by a label. We show that this is NP-hard in
general, but the problem can be solved in polynomial time if the road map is an
embedded tree.Comment: extended version of a CIAC 2015 pape
Optimal Interleaving on Tori
This paper studies -interleaving on two-dimensional tori. Interleaving has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. A -interleaving of a graph is defined as a vertex coloring in which any connected subgraph of or fewer vertices has a distinct color at every vertex. We say that a torus can be perfectly t-interleaved if its t-interleaving number (the minimum number of colors needed for a t-interleaving) meets the sphere-packing lower bound, . We show that a torus is perfectly t-interleavable if and only if its dimensions are both multiples of (if t is odd) or t (if t is even). The next natural question is how much bigger the t-interleaving number is for those tori that are not perfectly t-interleavable, and the most important contribution of this paper is to find an optimal interleaving for all sufficiently large tori, proving that when a torus is large enough in both dimensions, its t-interleaving number is at most just one more than the sphere-packing lower bound. We also obtain bounds on t-interleaving numbers for the cases where one or both dimensions are not large, thus completing a general characterization of t-interleaving numbers for two-dimensional tori. Each of our upper bounds is accompanied by an efficient t-interleaving scheme that constructively achieves the bound
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
Periodicity and Circle Packing in the Hyperbolic Plane
We prove that given a fixed radius , the set of isometry-invariant
probability measures supported on ``periodic'' radius -circle packings of
the hyperbolic plane is dense in the space of all isometry-invariant
probability measures on the space of radius -circle packings. By a periodic
packing, we mean one with cofinite symmetry group. As a corollary, we prove the
maximum density achieved by isometry-invariant probability measures on a space
of radius -packings of the hyperbolic plane is the supremum of densities of
periodic packings. We also show that the maximum density function varies
continuously with radius.Comment: 25 page
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