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We prove t h a t t h e crossing number of C4 X Ca is 8. 0 1995 John Wiley & Sons, Inc.It has long been conjectured that the crossing number of C, X C,, denoted u(C, X C n ) , is ( m -2)n for m I n . While ( m -2)n is easily seen to be an upper bound (see Figure l), equality has been established only for m = 3,4. Harary, Kainen, and Schwenk [3] proved that u(C3 X C,) = 3, and Beineke and Ringeisen [4] used this to prove that u(C3 X C,) = n for n 2 3 . Beineke and Ringeisen [ l ] also proved that v(C4 X C,) = 2n for n 2 4, but their proof relied on a 1970 result of Eggleton and Guy [2] that u(C4 X C4) = 8. The proof of the latter result has never been published. It is the purpose of this note to prove that u(C4 X C4) = 8.In what follows, all drawings will be assumed to be simple, i.e., no edge crosses itself, adjacent edges do not cross, crossing edges do so only once, edges do not cross vertices, and no more than two edges cross at a common point. A drawing of a graph G is optimal if it has the minimal number of possible crossings; this number is denoted u ( G ) . We note that C4 X Cq is isomorphic with the 4-cube Q4 and that it contains eight subgraphs isomorphic with the 3-cube Q 3 . Definition 1. edges of G that are incident with both X and V(G)\X.For a subset X of vertices of a graph G, 6 X is the set of Lemma 1. then 16x1 2 4.Journal of Graph Theory, Vol. 19, No. 1, 125-129 (1995) Let X be a subset of V = V(Q3). If 1x1 2 2 and IV\Xl 2 2, 0 1995 John Wiley & Sons, Inc.

Over the past twenty years, rectangle visibility graphs have generated considerable interest, in part due to their applicability to VLSI chip design. Here we study unit rectangle visibility graphs, with fixed dimension restrictions more closely modeling the constrained dimensions of gates and other circuit components in computer chip applications. A graph G is a unit rectangle visibility graph (URVG) if its vertices can be represented by closed unit squares in the plane with sides parallel to the axes and pairwise disjoint interiors, in such a way

The paper considers representations of bipartite graphs as rectanglevisibility graphs, i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. It is shown that, for p < q, Kp, q has a representation with no rectangles having collinear sides if and only ifp < 3 or p = 3 and q _~ 4. More generally, it is shown that Kp,q is a rectangle-visibility graph if and only if p < 4. Finally, it is shown that every bipartite rectangle-visibility graph on n > 4 vertices has at most 4n-12 edges.

Abstract. We study the problem of drawing a graph in the plane so that the vertices of the graph are rectangles that are aligned with the axes, and the edges of the graph are horizontal or vertical lines-of-sight. Such a drawing is useful, for example, when the vertices of the graph contain information that we wish displayed on the drawing; it is natural to write this information inside the rectangle corresponding to the vertex.We call a graph that can be drawn in this fashion a rectangle-visibility graph, or RVG. Our goal is to find classes of graphs that are RVGs.We obtain several results:1. For 1 < k < 4, k-trees are RVGs. 2. Any graph that can be decomposed into two caterpillar forests is an RVG. 3. Any graph whose vertices of degree four or more form a distance-two independent set is an RVG. 4. Any graph with maximum degree four is an RVG. Our proofs are constructive and yield linear-time layout algorithms.