Dan Pritikin

On the separation number of a graph

We consider the following graph labeling problem, introduced by Leung et al. (J. Y-T. Leung, O. Vornberger, and J. D. Witthoff, On some variants of the bandwidth minimization problem. SIAM J. Comput. 13 (1984) 650–667). Let G be a graph of order n, and f a bijection from V(G) to the integers 1 through n. Let |f|, and define s(G), the separation number of G, to be the maximum of |f| among all such bijections f. We first derive some basic relations between s(G) and other graph parameters. Using a general strategy for analyzing separation number in bipartite graphs, we obtain exact values for certain classes of forests and asymptotically optimal lower bounds for grids and hypercubes.

The harmonious coloring number of a graph

Abstract Hopcroft and Krishnamoorthy (1983) have shown that the harmonious coloring problem is NP-complete, introducing the notion of a harmonious coloring of a graph as being a vertex coloring for which no two edges receive the same color-pair. In this report we construct efficient harmonious colorings of complete binary trees, 2 and 3-dimensional grids, and n -dimensional cubes.

Applying a proof of tverberg to complete bipartite decompositions of digraphs and multigraphs

Graham and Pollak [2] proved that n – 1 is the minimum number of edge-disjoint complete bipartite subgraphs into which the edges of Kn decompose. Tverberg [6], using a linear algebraic technique, was the first to give a simple proof of this result. We apply Tverbergs technique to obtain results for two related decomposition problems, in which we wish to partition the arcs/edges of complete digraphs/multigraphs into a minimum number of arc/edge-disjoint complete bipartite subgraphs of appropriate natures. We obtain exact results for the digraph problem, which was posed by Lundgren and Maybee [4]. We also obtain exact results for the decomposition of complete multigraphs with exactly two edges connecting every pair of distinct vertices.

Expansion of layouts of complete binary trees into grids

Let Th be the complete binary tree of height h. Let M be the infinite grid graph with vertex set Z2, where two vertices (x1, y1) and (x2, y2) of M are adjacent if and only if |x1-x2| + |y1-y2|= 1. Suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of M. Motivated by issues in optimal VLSI design, we show that the point expansion ratio n(T)/n(Th) = n(T)/(2h+1 - 1) is bounded below by 1.122 for h sufficiently large. That is, we give bounds on how many vertices of degree 2 must be inserted along the edges of Th in order that the resulting tree can be laid out in the grid. Concerning the constructive end of VLSI design, suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of the n × n grid graph. Define the expansion ratio of such a layout to be n2/n(Th)=n2/(2h+1 - 1). We show constructively that the minimum possible expansion ratio over all layouts of Th is bounded above by 1.4656 for sufficiently large h. That is, we give efficient layouts of complete binary trees into square grids, making improvements upon the previous work of others. We also give bounds for the point expansion and expansion problems for layouts of Th into extended grids, i.e. grids with added diagonals.

Bounded dilation maps of hypercubes into Cayley graphs on the symmetric group

LetG andH be graphs with ¦V(G)≤ ¦V(H)¦. Iff:V(G) →V(H) is a one-to-one map, we letdilation(f) be the maximum of distH(fx),f(y)) over all edgesxy inG where distH denotes distance inH. The construction of maps fromG toH of small dilation is motivated by the problem of designing small slowdown simulations onH of algorithms that were originally designed for the networkG.LetS(n), thestar network of dimension n, be the graph whose vertices are the elements of the symmetric group of degreen, two verticesx andy being adjacent ifx o (1,i) =y for somei. That is,xy is an edge ifx andy are related by a transposition involving some fixed symbol (which we take to be 1). Also letP(n), thepancake network of dimension n, be the graph whose vertices are the elements of the symmetric group of degreen, two verticesx andy being adjacent if one can be obtained from the other by reversing some prefix. That is,xy is an edge ifx andy are related byx o (1,i(2,i-1) ⋯ ([i/2], [i/2]) =y. The star network (introduced in [AHK]) has nice symmetry properties, and its degree and diameter are sublogarithmic as functions of the number of vertices, making it compare favorably with the hypercube network. These advantages ofS(n) motivate the study of how well it can simulate other parallel computation networks, in particular, the hypercube.The concern of this paper is to construct low dilation maps of hypercube networks into star or pancake networks. Typically in such problems, there is a tradeoff between keeping the dilationsmall and simulating alarge hypercube. Our main result shows that at the cost ofO (1) dilation asn→ ∞, one can embed a hypercube of near optimum dimension into the star or pancake networksS(n) orP(n).More precisely, lettingQ (d) be the hypercube of dimensiond, our main theorem is stated below. For simplicity, we state it only in the special case when the star network dimension is a power of 2. A more general result (applying to star networks of arbitrary dimension) is obtained by a simple interpolation.

k-Dependence and Domination in Kings Graphs.

For most choices of k = 0, ... , 8, there is a tidy solution: an upper bound can be proved by a short elementary argument, and an arrangement of kings can be con structed to show that the upper bound is tight. These limiting densities are given in Section 6. However, tight upper bounds are not yet known for either k = 4 or k ? 5. It is easy to construct arrangements of kings (on arbitrarily large boards) that achieve the densities of 3/5 and 9/13 for k ? 4 and 5, respectively. We conjecture that these are indeed the maximum limiting densities. The story in the present article concerns the struggle to support this conjecture by good upper bounds, as well as the variety of rival techniques used for different val ues of k. Along the way, we make elementary use of graph theory, number theory, group theory, real analysis, and integer linear programming. We believe the methods of the present paper can provide the basis for undergraduate research projects on re lated problems.

All Unit-Distance Graphs of Order 6197 Are 6-Colorable

Consider any 6197 or fewer points in the plane, and create a graph with this vertex set by considering a pair of those points to be adjacent if and only if their distance is exactly 1. It is shown that the vertices of the resulting graph can be properly 6-colored.

On randomized greedy matchings

We analyze a randomized greedy matching algorithm (RGA) aimed at producing a matching with a large number of edges in a given weighted graph. RGA was first introduced and studied by Dyer and Frieze in [3] for unweighted graphs. In the weighted version, at each step a new edge is chosen from the remaining graph with probability proportional to its weight, and is added to the matching. The two vertices of the chosen edge are removed, and the step is repeated until there are no edges in the remaining graph. We analyze the expected size μ(G) of the number of edges in the output matching produced by RGA, when RGA is repeatedly applied to the same graph G. Let r(G) = μ(G)/m(G), where m(G) is the maximum number of edges in a matching in G.

Representation Numbers of Stars

Abstract A graph G has a representation modulo r if there exists an injective map ƒ : V(G) → {0, 1, . . . , r – 1} such that vertices u and 𝑣 are adjacent if and only if ƒ(u) – ƒ(𝑣) is relatively prime to r. The representation number rep(G) is the smallest positive integer r for which G has a representation modulo r. In this paper we study representation numbers of the stars K 1,n . We will show that the problem of determining rep(K 1,n ) is equivalent to determining the smallest even k for which φ(k) ≥ n: we will solve this problem for “small” n and determine the possible forms of rep(K 1,n ) for sufficiently large n.

Irredundance and domination in kings graphs

Each king on an n × n chessboard is said to attack its own square and its neighboring squares, i.e., the nine or fewer squares within one move of the king. A set of kings is said to form an irredundant set if each attacks a square attacked by no other king in the set. We prove that the maximum size of an irredundant set of kings is bounded between (n - 1)2/3 and n2/3, and that the minimum size of a maximal irredundant set of kings is bounded between n2/9 and ⌈(n + 2)/3⌉2, where the latter upper and lower bounds are in fact equal when n ≡ 0(mod3). Results are given for related domination and independence problems.