Zevi Miller

The biparticity of a graph

The biparticity β(G) of a graph G is the minimum number of bipartite graphs required to cover G. It is proved that for any graph G, β(G) = {log2χ(G)}. In view of the recent announcement of the Four Color Theorem, it follows that the biparticity of every planar graph is 2.

On graphs containing a given graph as center

We examine the problem of embedding a graph H as the center of a supergraph G, and we consider what properties one can restrict G to have. Letting A(H) denote the smallest difference ∣V(G)∣ - ∣V(H)∣ over graphs G having center isomorphic to H it is demonstrated that A(H) ≤ 4 for all H, and for 0 ≤ i ≤ 4 we characterize the class of trees T with A(T) = i. for n ≥ 2 and any graph H, we demonstrate a graph G with point and edge connectivity equal to n, with chromatic number X(G) = n + X(H), and whose center is isomorphic to H. Finally, if ∣V(H)∣ ≥ 9 and k ≥ ∣V(H)∣ + 1, then for n sufficiently large (with n even when k is odd) we can construct a k-regular graph on n vertices whose center is isomorphic to H.

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.

NP-Completeness for Minimizing Maximum Edge Length in Grid Embeddings

Abstract Given an embedding f: G → Z 2 of a graph G in the two-dimensional lattice, let |f| be the maximum L1 distance between points f(x) and f(y) where xy is an edge of G. Let B2(G) be the minimum |f| over all embeddings f. It is shown that the determination of B2(G) for arbitrary G is NP-complete. Essentially the same proof can be used in showing the NP-completeness of minimizing |f| over all embeddings f: G → Z n of G into the n-dimensional integer lattice for any fixed n ≥ 2.

Embedding grids into hypercubes

Abstract We consider efficient simulations of mesh connected networks (or good representations of array structures) by hypercube machines. In particular, we consider embedding a mesh or grid G into the smallest hypercube that at least as many points as G, called the optimal hypercube for G. In order to minimize simulation time we derive embeddings which minimize dilation, i.e., the maximum distance in the hypercube between images of adjacent points of G. Our results are: (1) There is a dilation 2 embedding of the [m × k] grid into its optimal hypercube, provided that ⌈ log m⌉+ log mk 2 ⌈ log m⌉ + log m 2 ⩽⌈ log mk⌉ and (2) For any k B k = a d ∏ k i=1 a i ∏ k i=1 2 ⌈ log a i ⌉ + ∑ i=1 k ⌊ log a 1 ⌋ 2 ⌉

Matroids and subset interconnection design

A problem arising in the design of vacuum systems and having applications to some natural problems of interconnection design is described as follows. (1) Given a set X and subsets

Expansion of layouts of complete binary trees into grids

X_i ,Y_i

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

of

Which trees are link graphs

X,i = 1, \cdots ,n