TILE WITH RATIONAL AVERAGE CROSSING NUMBER

TILE WITH RATIONAL AVERAGE

CROSSING NUMBER

Benny Pinontoan

Abstract. A tileT is a connected graph with two sequences of vertices, the left- and right-walls. The tile crossing numbertcr(T) of tileTis defined as the smallest number of crossings among all drawings ofTin a unit square where the vertices of left-wall and the vertices of the right-wall are drawn in the right order. LetTn

denote the big tile obtained by gluingncopies ofT in a linear fashion. The average crossing numberacr(T) of tile

Tis defined as lim

n→∞ tcr(Tn

)

n . In this paper, we show that average crossing number can

be a rational number.

1. INTRODUCTION

Bycrossing numbercr(G) of a graphGwe understand the minimal possible number of crossings among all its drawings on the plane [3]. Crossing number of a graph can be used to obtain a lower bound on the amount of chip area required by that graph in a VLSI circuit layout [4]. In general, crossing number is hard to compute; the crossing number problems are NP-hard [2].

Many well-known examples of infinite sequence of graphs, including the Gen-eralized Petersen GraphsP(n, k) (e.g. [8]) and the Cartesian productsCm×Cnof two cycles (e.g. [1]), are constructed by gluing many copies of a small piece, called thetile, in a circular fashion. The concept of tile was introduced in [6].

LetTn_and◦(Tn_{) denote the arrangements obtained by gluingncopies of tile}

T in linear and circular fashions, respectively. The average crossing numberacr(T)

of tile T is lim

Received 19 May 2005, Revised 1 August 2005, Accepted 18 August 2005.

2000 Mathematics Subject Classification: 05C10.

Key words and Phrases: tile, crossing number, tile crossing number, average crossing number.

Richter [7] asked the question whether average crossing number can be a rational number. This question is interesting in that it might motivate us to better understand how the average crossing number behaves, in order to understand what determines the average crossing number. In this paper, we answer this question by presenting a tile with average crossing number a half.

2. TILE AND AVERAGE CROSSING NUMBER

We recall first the formal definitions of tile and average crossing number. For more detail, we refer to [5, 6].

A tile is a 3-tupleT = (G, L, R), where

• Gis a connected graph;

• L = L[1], L[2], ..., L[|L|] is a finite sequence of vertices of V(G), called the left-wall;

• R =R[1], R[2], ..., R[|R|] is a finite sequence of vertices of V(G), called the right-wall; and

• all the vertices inLandRare distinct.

Consider the square S _{= {(x, y) : −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1}. A tile}

drawingof a tileT = (G, L, R) is a drawing ofT inS_{such that: (i) the intersection}

with the boundary{(x, y)∈S_{: eitherx∈ {−1,1}ory∈ {−1,1}}is preciselyL∪R;}

(ii) the vertices in left-wall Loccur in the line x=−1, with the y-coordinates of

L[1], L[2], ..., L[|L|] decreasing; and (iii) the vertices in right-wall R occur in the line x= 1, with they-coordinates ofR[1], R[2], ..., R[|R|] also decreasing. The tile crossing number tcr(T) of a tileT is the smallest number of crossings in any tile drawing ofT.

A tile T1 = (G1, L1, R1) is compatible with a tile T2 = (G2, L2, R2) if the functionf :R1→L2defined byf(R1[i]) =L2[i] is bijection and

∀v1, v2∈R1:v1v2∈E(G1)⇔f(v1)f(v2)∈E(G2).

A tile isself-compatible if it is compatible to itself. Note that the compatibility of tiles refers to ordered pairs of tiles.

LetGandH be (disjoint) graphs and let X andY be sequences of distinct vertices ofGandH, respectively, having the same lengtht. Then (G∪H){X, Y}is the graph obtained by identifyingX[i] withY[i], fori= 1,2, ..., t. LetS= (G, L, Q) andT = (H, M, R) be compatible tiles. ThenST is the tile obtained by identifying the right-wall ofS with the left-wall of T, that is, ST = ((G∪H){Q, M}, L, R). If T = (G, L, R) is a self-compatible tile, then the circular arrangement ◦(T) of tileT isG{L, R} and the linear arrangementTn_{of tileT is defined inductively by}

LetT be a tile. Then we define theaverage crossing number acr(T) ofT to

be lim n→∞

tcr(Tn₎

n .

Theorem 1. (Pinontoan and Richter [5])Let T be a tile. Thenacr(T)exists and

equals to inf

For more properties of average crossing number, we refer to [5].

3. TILE WITH RATIONAL AVERAGE CROSSING NUMBER

This section presents the main result of this paper, namely to show that average crossing number of tile can be a rational number.

Theorem 2. There exists a tile with a rational average crossing number.

Proof. We recall that a graph isouterplanar if it can be embedded (without cross-ing) in the plane so that every vertex of that lies on the unbounded face. A graph is an outerplanar if and only if it contains no subgraph that is a subdivision of either

K4 orK2,3.

so it is not outerplanar. There are only three possible drawings of H_{i with no}

crossings among the edges of H_{i, namely, drawing}D_{1 with ei andxi+1 inside the}

cycleC1 =aibifiei+1di+1diai, drawingD2with ai andbi inside the cycleC2 =

dieixi+1fiei+1di+1di and drawingD3 withdi+1 andei+1 inside the cycle C3 = aibifixi+1eidiai.

We claim that in each of these three possible drawings, we can assign one crossing toH_{i. If any two edges of}H_{icross, then we count the crossing as one and}

assign this crossing toH_{i. In drawing}D_{1, there is a path fromei to (a vertex of)}

Land another path fromeito (a vertex of)R. Each of these paths is disjoint from

C1 and so they both must cross that cycle. We count each of the crossings a half and so we have one crossing assigned forH_i.

Note thatD_{2 is only possible when i≥2. For this drawing, there is a path}

fromai toa1 and another path frombi to eitherb1 orc1. These paths are disjoint from each other and from C2 and so each of them must crossC2. We count each of these crossings a half and so we have one full crossing to be assigned toH_i.

Similarly, drawingD_{3is only possible wheni≤n−2. For this drawing, there}

a

paths are disjoint from each other and from C3 and so each of them must cross

C3. We count each of these crossings a half and so we have one full crossing to be assigned toH_i.

Thus, for each of the possible drawings ofH_{i, we can assign one full crossing}

to it. Note that for each suchH_{i, there is only one common edge with its adjacent}

neighbor (H_{i−2 or}H_{i+2), but an edge cannot cross itself. Hencetcr(}Tn_)≥ n

Acknowledgement. We thank R.B. Richter for suggesting the problem and useful discussions. We thank the referee(s) for useful suggestions to improve this paper.

REFERENCES

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crossing number ofC7×Cn”,J. Combin. Theory Ser. B90(2004), 21–39.

2. M.R. Garey and D.S. Johnson_{, “Crossing number is NP-complete”,}SIAM J. Alg. Disc. Meth. 4_{(3) (1983), 312–316.}

3. P. Erd¨os and R.K. Guy_{, “Crossing number problems”,} Amer. Math. Monthly80 (1973), 52–58.

4. F.T. Leighton, “New lower bound techniques for VLSI”,Proc. 22nd Annual Sympo-sium on Foundation of Computer Science, IEEE Computer Society, Long Beach, CA, (1981), 1–12.

5. B. Pinontoan and R.B. Richter, “Crossing numbers of sequences of graphs I: Gen-eral tiles”,Australas. J. Combin. 30(2004), 197–206.

6. B. Pinontoan and R.B. Richter, “Crossing numbers of sequences of graphs II: Planar tiles”,J. Graph Theory42(2003), 332–341.

8. R.B. Richter and G. Salazar_{, “The crossing number ofP(n,3)”,}Graphs Combin. 18(2002), 381–394.

B. Pinontoan: Department of Mathematics, Sam Ratulangi University, Manado 95115,

Indonesia.