AND COMPLETE MULTIPARTITE GRAPHS Yi-Wu Chang
National Politics University, Taipei, Taiwan
Douglas B. West y
University of Illinois, Urbana, IL 61801-2975, west@math.uiuc.edu
Abstract. The rectangle number of a graph G is the minimum t such that G is the intersection graph of sets that are unions of t rectangles in the plane with vertical and horizontal sides. We prove that complete multipartite graphs have rectangle number at most two, and that the k-dimensional hypercube has rect-angle number at most dk=4e (except one more whenk = 4).
1. INTRODUCTION
The intersection graph of a family of sets fS 1
:::Sng is the graph with vertex set fv
1
:::vng de ned by making vi and vj adjacent precisely when Xi \Sj 6= . The sets in the family form anintersection representation of the corresponding graph. The interval graphs are those having intersection representations in which each set is an interval on the real line this special family has been thoroughly studied in hundreds of papers.
Sets generalizing intervals have been used to permit intersection representations of all graphs. Natural parameters measure how much the sets deviate from being intervals. A d-box in Rdis a cartesian product of d intervals. A t-interval in R
1 is a union of
t inter-vals. The boxicity box(G) of a graph G is the minimumd such that G is the intersection graph of a family of d-boxes. The interval number i(G) is the minimumt such that G is the intersection graph of a family of t-intervals. Boxicity was introduced by Roberts 3] interval number by Trotter and Harary 7].
The ideas behind these two parameters can be combined. In any xed dimension d, we can seek the minimum t such that G is the intersection graph of at most t d-boxes. When d = 2, the resulting parameter is the rectangle number r(G). In this paper, we compute the rectangle number for all complete multipartite graphs (it is always 2 or 1), y
Research supported in part by NSA/MSP Grant MDA904-93-H-3040. Running head: RECTANGLE NUMBER
AMS codes: 05C35
Keywords: intersection graph, rectangle number, representation parameter, hypercube, complete multipartite graph
2
and we derive an upper bound on the rectangle number of the k-dimensional hypercube (it is at most dk=4e, except that it equals 2 when k = 4).
The result for complete multipartite graphs is striking in relation to their interval number and boxicity. The full spectrum of box intersection parameters starts with the interval number and ends with 1 when the dimension equals the boxicity. Trotter and Harary 7] proved that i(Kmn) = dmn
+1
m+n
e. Hopkins, Trotter, and West 2] proved that i(G) =dmn
+1
m+n
e when G is a complete multipartite graph in which the two largest partite sets induce Kmn, except in rare special cases where it is larger by 1.
Roberts 3] proved that the boxicity of a complete multipartite graph equals the num-ber of partite sets with size at least 2. The boxicity can be interpreted as the numnum-ber of dimensions that must be allowed for the \multiplicity" of representation to decline from dmn
+1
m+n
eto 1. Surprisingly, almost all of the collapse in multiplicity occurs on the rst step: the rectangle number of every complete multipartite graph is at most 2.
Best-possible bounds on the full spectrum are known for the family of planar graphs. For every planar graph, the interval number is at most 3 5], the rectangle number is at most 2 4], and the boxicity is at most 3 6]. Thus in each dimension, the dimension plus the multiplicity is at most 4, and these bounds are sharp.
For general n-vertex graphs, the interval number is at most d(n+ 1)=4e 1], and the boxicity is at mostdn=2e 3]. We do not know the maximum rectangle number forn-vertex graph or any general upper or lower bounds on how the maximumd-box intersection num-ber for n-vertex graphs declines as d increases from 1 todn=2e. Even for the hypercubes, we have not proved a general lower bound, but we believe that the rectangle number of hypercubes increases linearly with the dimension, and thus that the maximum rectangle number increases at least logarithmically with the number of vertices.
When we speak of the horizontal projection or vertical projection of a rectangle, we mean its projection on the horizontal or vertical axis, respectively. In other words, it is the rst or second factor, respectively, in the description of the rectangle as a cartesian product of intervals.
2. COMPLETE MULTIPARTITE GRAPHS
THEOREM 1. If G is a complete multipartite graph, then r(G) 2, with equality if and only if G has at least three partite sets of size at least 2.
Proof: If G does not have at least three partite sets of size at least 2, then by Roberts' result 3] the boxicity of G is at most 2, which allows G to be represented using one rect-angle per vertex. Indeed, the vertices of one nontrivial partite set can be represented by parallel thin vertical strips, the other vertices by parallel thin horizontal strips, and the vertices in partite sets of size 1 by rectangle that intersect everything.
Conversely, suppose that r(G) = 1. Let H be an copy of C 4 =
K
22 as an induced subgraph in G, with partite sets X and Y. Since X and Y induce K
2, we may assume by symmetry that the rectangles for X have disjoint horizontal projections. The horizon-tal projections for Y must both contain the gap between these and therefore intersect. Thus the rectangles for Y have disjoint vertical projections, and the vertical projections forX intersect. If G contains K
222, then the three pairwise edge-disjoint copies of C
3
this subgraph cannot all satisfy this. Two of the three pairs of independent points have rectangles with disjoint projections in the same direction thus these cannot representC
4. Finally, we construct a rectangle representation with two rectangles per vertex forG. Consider a block adjacency matrix for G with the vertices grouped by partite sets. We obtain a representation from the portion of this matrix above the diagonal. We view the ones in each row as a thin horizontal rectangle for that vertex. We view the ones in each column as a thin vertical rectangle for that vertex. Each partite set is represented by two families of thin parallel rectangles. The rectangles do not extend as far as the \diagonal", so they intersect if and only if the corresponding vertices belong to distinct partite sets. Fig. 1 illustrates the construction.
5 4 3 6 5 4 2 1 3 2
Fig. 1. Rectangle representation for complete multipartite graph.
3. HYPERCUBES
The 4-dimensional hypercube is an annoying complication in our inductive construc-tion of rectangle representaconstruc-tions for hypercubes. The inducconstruc-tion step is quite easy, but r(Q
4) = 2 requires us to provide explicit representations for Q
8 and Q
12. A
t-rectangle
representation is an intersection representation assigning each vertex the union of at most t rectangles in the plane (with horizontal and vertical sides). Themultiplicity of a vertex in a representation is the number of rectangles assigned to it.
In the k-dimensional hypercube Qk, we view the vertices as subsets of f1:::kg adjacent when their subset labels di er by one element. We begin by analyzing Q
4. LEMMA 2. In every rectangle representation of Q
4, at least two vertices have multi-plicity at least two.
Proof: Let f be a rectangle representation of Q
4. We prove that within distance 3 of an arbitrary vertex v in Q
4, there is a vertex with multiplicity at least two. Hence some vertex has multiplicity at least two, and applying the argument again to the complement of that vertex yields a second such vertex.
Without loss of generality, let v = , and suppose that every vertex other than f1234ghas multiplicity one. LetRudenote the unique rectangle assigned tou(dropping set brackets). Because Q
4 is triangle-free, no assigned rectangle is contained in another. Also, the rectangle R for v intersects four pairwise disjoint rectangles R
1 R 2 R 3 R 4 for its neighbors. For ij 2 f1234g, we claim that Ri and Rj intersect a common side of R. If not, then Rij will also intersect R, as shown in Fig. 2.
4 R Ri Rj R Ri Rj
Fig. 2. Restrictions on neighbors in representation of Q 4. Without loss of generality, we may assume that R
1 R 2 R 3 R
4 intersect the left side of R in that order from top to bottom (they may also extend past the right side of R). Since R
13 extends vertically from R
1 to R
3 but misses R
2, it must lie entirely to the left or right of R by symmetry, we may assume that it lies to the left. Now R
13 blocks the leftward advance of R
2, and R
3 extends farther left than R
2. Hence R
4 must be to the right of R and block the rightward advance of R
3. Next, R
23 must extend across the vertical gap between R 2 and R 3 but avoid R. Hence R
23 is entirely to the left or to the right of
R we may assume by symmetry that it lies to the right. Fig. 3 displays the conclusions that will follow from this.
1 2 3 4 24 23 12 13 14 123 124
Fig. 3. Arrangement of singleton rectangles in representation ofQ 4. Since R 23 must meet R 3 and avoid R
24, the horizontal extent of R 23 is now con ned between R and R 24. Since R 23 must meet R 3 and avoid R
4, the vertical extent of R 23 is bounded below byR 4. Since R 123 must meet R 13 and R 23 and avoid R, and since R 23 is bounded below by R 4, we conclude that R 123 extends from R 13 to R 23 above R. Hence R
23 blocks the rightward advance of R
1. Since R
14 extends vertically from R
1 to R
4, and R
2 extends past the right end of R
1 between them, R
14 must lie to the left of
R. Indeed, it must also lie to the left of R
3. Now, R
12 must extend vertically from R
2 to R
1 and must avoid R
13 and R
5 the horizontal extent of R
12 is bounded between R
13 and R
23. Furthermore, the vertical extent of R 12 is bounded below by R 3. Since R 12 and
R each extend vertically between R
1 and R
2, their horizontal projections are disjoint either may be leftmost. Next, R
124 must extend horizontally between R 14 and R 24, meeting R 12 and avoiding R 13 and R 23. Since R 12 lies above R 3, we conclude that R 124 lies above R
123, and that all of R
14 R
12 R
24 extend up far enough to meet it.
We have now located all regions except those whose indices contain both 3 and 4. BothR 13 and R 23 lie above R 4. Hence R 134and R
234, which must avoid R
4, also lie above R
4. Since R
34 extends vertically between R
3 and R
4 and avoids
R, it lies entirely to the left or to the right of R. If R
34 lies to the left right] of
R, then R
234 cannot meet both R 34 and R 23 R 13] without crossing R, since R 34 lies below R 1 R 2]. We have proved that there is no rectangle representation of Q
4 in which every vertex within distance three of a speci ed vertex has multiplicity 1.
LEMMA 3. There is a 2-rectangle representation of Q
4 in which the only vertices with multiplicity 2 are two adjacent verticesxy, also one rectangle forxemerges vertically in both directions (arbitrarily far) and the other emerges vertically in one direction, and the rectangles for y satisfy the same behavior horizontally.
Proof: By allowing two rectangles for each of 234 and 1234, we can complete the repre-sentation begun in Fig. 3 as indicated in Fig. 4. These four rectangles can be extended so that the rectangles for 1234 both extend upward and one extends downward, and the rectangles for 234 both extend rightward and one extends leftward.
1 2 3 4 24 23 12 13 14 123 124 1234 34 134 234 234 1234
Fig. 4. The canonical 2-rectangle representation of Q 4.
6
By symmetry, we may choose any adjacent pair of vertices to receive two rectangles each, and we may exchange up/down and right/left at will in this representation. Call this the canonical representation of Q
4.
For the main result, we view the vertices of Qk as binary vectors of length k.
THEOREM 4. The rectangle number of the k-dimensional cube Qk is at most dk=4e, except that r(Q
4) = 2. Proof: We have proved that r(Q
4) = 2. Since Q
4 contains a copy of Q
3 avoiding a pair of adjacent vertices, the canonical representation contains a 1-rectangle representation of Q
3. Since
Qk Qk
+1 for all
k, it therefore su ces to prove that r(Q 4l
l for eachl >1. We use induction onl. Forl>1, we viewQ
4las Q
8 Q
4l;8, where denotes cartesian product of graphs. For each choice of bits in the last 4l;8 coordinates of vertices ofQ
4l, we have a copy of Q
8. If Q
8 has a 2-rectangle representation, then the disjoint union of 24l;8 copies of
Q
8 has a 2-rectangle representation. Deleting the edges of these copies of Q
8 leaves 8 disjoint copies of Q
4l;8. If
l 4, then the induction hypothesis allows us to complete the representation using l;2 additional rectangles per vertex.
To complete the proof, we provide constructions for r(Q
8) = 2 and r(Q
12)
3. We use the canonical representation of Q
4 provided by Lemma 3. 2-rectangle representation of Q 8. Express Q 8 as Q 4 Q 4. View Q 8 as G H, where each ofGH is the disjoint union of 16 copies ofQ
4. Each vertex appears in one component in each ofGandH. Each component ofG resp.,H] has a xed value in the last four rst four] coordinates of its vertex labels. Our representation has eight isomorphic connected
pieces, each of which represents two components of G and two components ofH.
Instead of describing an explicit representation, we describe a class of representa-tions, because we will use eight di erent 2-rectangle representations of Q
8 to construct a 3-rectangle representation of Q
12. We parse each 8-bit label of a vertex of Q
8 as a concate-nation , where are single bits and are 3-bit binary vectors. We use vector addition modulo 2. In the 8-bit label of v, we refer to the vector in coordinates 2-4 as (v) and the vector in coordinates 6-8 as (v).
Given a xed 3-bit vector z, we describe a representation of Q
8 the eight choices for z yield eight di erent representations, with z as a parameter. Given a 3-bit binary vector x, let y = x+z (modulo 2), and let abcd respectively denote the four special vertices 0x0y1x0y1x1y0x1y. The subgraph induced by fabcdg is a 4-cycle consisting of one edge from each of two components of Gand one edge from each of two components of H. We label these four copies of Q
4 (components in
Gand H) asABBC CD D A such that ab2AB G, bc2BC H, cd2CDG, and da2D A H.
We represent AB BC CD D A using two rectangles each for abcd and one rectangle for the remaining vertices. No vertex outside fabcdg appears in more than one of these four subgraphs. We use four copies of the canonical representation of Q
4, extending the two rectangles for one of fabcdg out vertically in one direction and ex-tending the two rectangles for the appropriate neighbor amongfabcdg out horizontally in one direction. The two parallel rectangles fora emerging from the representation ofD A serve also as the two parallel rectangles for A emerging from the representation of AB, and similarly for bcd. The resulting representation is illustrated in Fig. 5.
7 BC CD D A c c a b b d AB a d
Fig. 5. A piece in the representation of Q 8.
For each of the eight choices for x, we have such a piece in our representationfz. A vertex v is a special vertex (abcd) in some piece if and only if (v) =(v) +z (modulo 2). A special vertex v appears only in the piece generated by x= (v), and it is assigned two rectangles in representing that component.
A non-special vertex u appears in the piece generated by x = (u) and also in the pieces generated by x = (u);z, assigned one rectangle in each piece. Thus we have represented the 32 pairwise edge-disjoint copies of Q
4 in eight disjoint pieces, obtaining a 2-rectangle representation of Q 8. 3-rectangle representation of Q 12. Express Q 12 as Q 8 Q
4. We represent the 16 copies of Q
8 in the manner described above in total we have 128 pieces. These pieces contain canonical representations of Q
4 in which we have made extra use of the parallel \tails" emerging in common directions. To avoiding adding two intervals for vertices in the re-maining 256 disjoint copies of Q
4, we will also make extra use of the third tail emerging in the opposite direction. Each of the 128 pieces has 8 tails emerging (we will use one for each of the four special vertices), and each of the 256 copies of Q
4 needs to use 2 of these tails. We produce a representation in connectedmodules, each containing two of the previously-constructed pieces.
We parse each 12-bit vertex label in Q
12 as a concatenation
, where are single bits and are 3-bit binary vectors. In the 12-bit label of v, we refer to the vector in coordinates 2-4 as (v), in coordinates 6-8 as (v), and in coordinates 10-12 as (v).
Our copies of Q 8 in
Q
12 have vertex labels xed in the last four coordinates. Using the ninth coordinate, we view these copies of Q
8 in eight pairs. In the pair for which the last three coordinates are xed at (v) = z, we use the representation fz for each of the two copies ofQ
8. We combine corresponding pieces of these two representations (and with the pair pick up four of the remaining copies ofQ
4) to produce 8
8 = 64 modules for our full representation.
Let abcd be the four special vertices in one piece of the representation fz on the copy of Q
8 having 0
z in the last four coordinates. Let a 0 b 0 c 0 d 0 be the corresponding vertices having 1 in the ninth coordinate. Note that w and w
0 are adjacent, for w 2 fabcdg. Each of these eight special vertices appears in one un-represented copy of Q
8
in which the rst eight coordinates are xed. Bothw and w
0 appear (and are adjacent) in a single such subgraph, which we call F(w).
We represent F(w) using the canonical representation, letting w w
0 be the adjacent pair using two rectangles each. Since we already have used two rectangles for each of w w
0 in the pieces of
fz, we can a ord only one additional rectangle. This we achieve for each of w w
0 by extending a rectangle from a piece of
fz to become a rectangle in the representation of F(w) (one horizontal, one vertical). The explicit geometric arrangement appears in Fig. 6. For xedz, let f
0
z denote the eight modules formed in this way.
BC CD D A c c a b b d AB a d BC 0 AB 0 D A 0 b 0 b 0 d 0 c 0 c 0 a 0 CD 0 d 0 a 0 F(b) b 0 b F(d) d 0 d F(c) c c 0 F(a) a a 0
Fig. 6. A module in the representation of Q 12. We have one pair of Q
8's for each choice of
z. For each special vertex w in each module of f
0
z, we have (w) = (w) + z and (w) = z. For each non-special vertex v that belongs to some F(w) in some module of f
0
z, we have (v) = (v) +z but (v)6= z. Thus a vertex v of Q
12 occurs as a special vertex in some module of some f
0
z if and only
if (v) =(v) + (v).
If (v) = (v) + (v), then v occurs as a special vertex in f 0
(v). The edges incident to v via changes in the rst eight coordinates occur in an f
(v)-piece in an f
0
(v)-module, and the edges via changes in the last four coordinates occur in the F(v)-portion of the f
0
(v)-module (or
F(w) if v = w
0), which assigns a third rectangle to
v. Furthermore, no rectangles for v appear in f
0
z with z 6= (v).
If (v) 6= (v) + (v), then v never occurs as a special vertex. In the pair of Q 8's corresponding to z = (v);(v) and represented by f
0
z, there are two intervals assigned
9 Also, vreceives one more interval in f
0
z, taking care of its incident edges via changes in the
last four coordinates. This rectangle appears in F(w) where w agrees with v in the rst eight or nine coordinates and is a special vertex.
We have veri ed that the union of the eight modules of the formf 0
z is a 3-rectangle
representation of Q 12.
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3] F.S. Roberts, On the boxicity and cubicity of a graph. InRecent Progress in Combina-torics (Proc. Third Waterloo Conf. on CombinaCombina-torics, 1968) (Academic Press, 1969), 301{310.
4] E.R. Scheinerman,Intersection classes and multiple intersection parameters of graphs. Ph.D. thesis, Princeton University, 1984.
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