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On the connectedness and diameter of a geometric Johnson graph
Crevel Bautista-Santiago, Javier Cano, Ruy Fabila-Monroy, David Flores-Peñaloza, Hernàn González-Aguilar, Dolores Lara, Eliseo Sarmiento,
Jorge Urrutia
To cite this version:
Crevel Bautista-Santiago, Javier Cano, Ruy Fabila-Monroy, David Flores-Peñaloza, Hernàn González-
Aguilar, et al.. On the connectedness and diameter of a geometric Johnson graph. Discrete Mathe-
matics and Theoretical Computer Science, DMTCS, 2013, Vol. 15 no. 3 (3), pp.21–30. �hal-00966378�
Discrete Mathematics and Theoretical Computer Science DMTCS vol.15:3, 2013, 21–30
On the Connectedness and Diameter of a Geometric Johnson Graph
C. Bautista-Santiago
1J. Cano
2R. Fabila-Monroy
3D. Flores-Pe˜naloza
4H. Gonz´alez-Aguilar
5D. Lara
6†E. Sarmiento
7J. Urrutia
21Dep. de Sistemas, Universidad Aut´onoma Metropolitana, Unidad Azcapotzalco, Mexico.
2Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Mexico.
3Dep. de Matem´aticas, Centro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico Nacional, Mexico.
4Dep. de Matem´aticas, Facultad de Ciencias, Universidad Nacional Aut´onoma de M´exico, Mexico.
5Facultad de Ciencias, Universidad Aut´onoma de San Luis Potos´ım Mexico.
6Dep. de Matem`atica Aplicada II, Universitat Polit`ecnica de Catalunya, Spain.
7Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Mexico
received 15thFeb. 2012,revised 3rdMay 2013,accepted 26thAug. 2013.
LetPbe a set ofnpoints in general position in the plane. A subsetIofPis called anislandif there exists a convex setCsuch thatI=P∩C. In this paper we define thegeneralized island Johnson graphofP as the graph whose vertex consists of all islands ofPof cardinalityk, two of which are adjacent if their intersection consists of exactlyl elements. We show that for large enough values ofn, this graph is connected, and give upper and lower bounds on its diameter.
Keywords:Johnson graph, intersection graph, diameter, connectedness, islands.
1 Introduction
Let[n] :={1,2, . . . , n}and letk≤nbe a positive integer. Ak-subset of a set is a subset ofkelements.
TheJohnson graphJ(n, k)is the graph whose vertex set consists of allk-subsets of[n], two of which are adjacent if their intersection has sizek−1. TheKneser graphK(n, k)is the graph whose vertex set consists of allk-subsets of[n], two of which are adjacent if they are disjoint. Thegeneralized Johnson graphGJ(n, k, l)is the graph whose vertex set consists of allk-subsets of[n], two of which are adjacent if they have exactlylelements in common. ThusGJ(n, k, k−1) =J(n, k)andGJ(n, k,0) =K(n, k).
Johnson graphs have been widely studied in the literature. This is in part for their applications in Network Design–where connectivity and diameter (i) are of importance. (Johnson graphs have small
†Email:maria.dolores.lara@upc.edu
(i)A graph is connected if there is a path between any pair of its vertices. The distance between two vertices is the length of the shortest path joining them. The diameter is the maximum distance between every pair of vertices of a graph.
1365–8050 c2013 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
diameter and high connectivity.) Geometric versions of these graphs have been defined in the literature.
In Araujo et al. (2005) the chromatic numbers of some “geometric type Kneser graphs” were studied. In this paper we study the connectedness and diameter of a “geometric” version of the generalized Johnson graph.
LetP be a set ofnpoints in the plane. A subsetI⊂Pis called an island if there exists a convex setC such thatI =P∩C. We say thatIis ak-island if it has cardinalityk(see Figure1). Let0≤l < k≤n be integers. Thegeneralized island Johnson graphIJ(P, k, l)is the graph whose vertex set consists of all k-islands ofP, two of which are adjacent if their intersection has exactlylelements. Note thatIJ(P, k, l) is an induced subgraph ofGJ(n, k, l). IfP is in convex position, thenIJ(P, k, l)andGJ(n, k, l)are isomorphic–since in this case every subset ofkpoints is ak-island.
Fig. 1:A subset of 5 points which is a 5-island, and a subset of 5 points which is not (both painted black).
Graph parameters ofIJ(P, k, l)can be translated to problems in Combinatorial Geometry of point sets.
Here are some examples.
- The number of vertices of this graph is the number ofk-islands ofP–the problem of estimating this number was recently studied in Fabila-Monroy and Huemer (2011).
- An empty triangle of P is a triangle with vertices onP and without points of P in its interior.
The empty triangles of P are precisely its 3-islands (or the number of vertices in IJ(P,3, l)).
Counting them has been a widely studied problem B´ar´any and F¨uredi (1987); B´ar´any and Valtr (2004); Dumitrescu (2000); Katchalski and Meir (1988); Valtr (1995).
- A related question B´ar´any and K´arolyi (2001) is: What is the maximum number of empty trian- gles that can share an edge? This translates to the problem of determining the clique number of IJ(P,3,2).
The paper is organized as follows. In Section 2, we prove thatIJ(P, k, l)is connected whennis large enough with respect tokandl. The proof of this result implies an upper bound ofO
n k−l
+O(k−l) on the diameter of this graph. In Section 3, we improve this bound for the case whenl ≤k/2, where we show that the diameter is at mostO(logn)+O(k−l). We also exhibit a choice ofPfor whichIJ(P, k, l) has diameter at leastΩ
logn−logk log(k−l)
. Note that these bounds are asymptotically tight whenl≤k/2and,
On the Connectedness and Diameter of a Geometric Johnson Graph 23
p0
p1
p2
p3
p4
p5
p6 p7
p8 p9
p10 p11
(a) Projectable.
p0
p1
p2
p3
p4
p5
p6 p7
p8 p9
p10 p11
(b) Non projectable.
Fig. 2:A4-island which is projectable, and a4-island which is not projectable.
kandlare constant with respect ton. A preliminary version of this paper appeared in Bautista-Santiago et al. (2010).
2 Connectedness
In this section we prove the following theorem.
Theorem 2.1 Ifn >(k−l)(k−l+ 1) +k, thenIJ(P, k, l)is connected and its diameter isO
n k−l
+
O(k−l).
The proof is divided in two parts:
• First we chooseF, an appropriate ‘core’ subgraph fromIJ(P, k, l)which has small diameter and is connected.
• Next we prove that for every vertex inIJ(P, k, l)there is a path of length at mostO
n k−l
con- necting it to a vertex inF.
2.1 Choosing a core subgraph from IJ(P, k, l)
LetP :={p0, p1, . . . , pn−1}be a set ofnpoints in general position in the plane. So thatp0is the topmost point ofP, andp1, . . . , pn−1are sorted counterclockwise by angle aroundp0. For0 ≤i ≤j ≤n, let Pi,j:={pi, pi+1, . . . , pj}and letPi,j′ :=Pi,j∪ {p0}. Observe thatPi,jandPi,j′ are both islands ofP. We call these two types of islandsprojectable, and define this concept next.
Definition 2.2 The projection of a pointpi,i6= 0, is the intersection of the ray emanating fromp0and passing throughpiwith a horizontal line. The projection of an island is the projection of its point set after omittingp0. Projectable islands are those islands that can be “projected” in such a way that its points are consecutive in the image of the whole set; see Figure 2.
LetFbe the subgraph ofIJ(P, k, l)induced by the projectablek-islands ofP. LetSbe a set ofn−1 points on a horizontal lineh, and letS′:=S∪ {x}, wherexis a point not inh. It is not hard to see thatF is isomorphic toIJ(S′, k, l). We classify the islands ofS′into two types: those that containx, and those that do not. Notice that these two types correspond to the two types of projectable islands ofP.
Now we show thatIJ(S′, k, l)is connected. First we consider the subgraph ofIJ(S′, k, l)induced by those islands ofS′that do not containx. Note that this is preciselyIJ(S, k, l). Without loss of generality assume thatSis a setx1< x2<· · ·< xn−1of points on the real line. Observe that ak-island ofSis an intervalofkconsecutive elements{xi, . . . , xi+k−1}. For the sake of clarity, in what follows we refer to k-islands ofSask-intervals.
Twok-intervals ofSare adjacent inIJ(S, k, l)if they overlap in exactlylelements. It follows easily that if l > 0, each k-interval is adjacent to at most two differentk-intervals, one containing its first element, and the other containing its last; see Figure 3. SinceIJ(S, k, l)has no cycles and its maximum degree is at most two, it is a union of pairwise disjoint paths. These paths can be described as follows. For i < j, letAiandAjbe the intervals ending atxiandxjrespectively. There is a path betweenAiandAj
if and only ifi≡j mod (k−l). Consider the interval adjacent withAito its right, this interval must end at pointxi+(k−l)(leaving exactlylpoints on the intersection). On the other hand, the interval adjacent withAito its left, must end at pointxi−(k−l); see Figure 3. For0≤r < k−l, letPrbe the subgraph of IJ(S, k, l)induced by thosek-intervals ending at a point with index congruent tor mod (k−l). Thus we have:
Proposition 2.3 If l > 0, Pr is an induced path of IJ(S, k, l). MoreoverIJ(S, k, l)is the union of {Pr|0≤r < k−l}.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A7 A11 A15
Fig. 3:Three differentk-intervals, for|S|= 16,k= 6,l= 2.
Forl = 0andn≥3k−1, everyk-interval would either intersect the left-mostk-interval or the right- mostk-interval, but not both. In this caseIJ(S, k,0)is connected and its diameter is at most3. Note that except for some special cases,IJ(S, k, l)is disconnected. Remarkably, as we show next, for a large enough value ofn, the addition of one extra point makes the graph connected.
As beforeAi is thek-island (k-interval) that ends at pointxi and does not containx. LetA′ibe the k-island ending at pointxi and containingx; see Figure 4. The structure ofIJ(S′, k, l)whenl < 2is different from whenl ≥2. In what follows, we assume thatl ≥2and briefly discuss the casel <2at the end of this section. Note that the subgraph ofIJ(S′, k, l)induced by theAi’s is preciselyIJ(S, k, l);
and the subgraph induced by the islandsA′iis isomorphic toIJ(S, k−1, l−1). From these observations, the following lemma is not hard to prove:
Lemma 2.4 Ifl≥2, then inIJ(S′, k, l):
1. Aiis adjacent toA′i−(k−l)andA′i+(k−l)−1(if they exist).
2. A′iis adjacent toAi+(k−l)andAi−(k−l)+1(if they exist).
The following theorem provides sufficient and necessary conditions forIJ(S′, k, l)to be connected.
Theorem 2.5 Forl≥2, the graphIJ(S′, k, l)is connected if and only ifn≥3k−2l−1orn=k.
On the Connectedness and Diameter of a Geometric Johnson Graph 25
A7 A15
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A′7 A′15
Fig. 4:Four differentk-islands inS′, for|S′|= 16,k= 6,l= 2.
Proof:LetIandJ be twok-islands ofS′. As long as the intermediate islands exists we can repeatedly use Lemma 2.4 to find a path fromIto an island whose endpoint is in the same residue class of(k−l)as the endpoint ofJ, and that containsxif and only ifJdoes. This is the case whenevern≥3k−2l−1:
Consider the set of islands {Ak, Ak+1, Ak+2, . . . , Ak+(k−l−1)}. Notice that, for any value of k and l ≥ 2, all islands in this set exist. Furthermore, there is exactly one island in the set for each residue class. Afterwards Proposition 2.3 ensures that there is a path from this island toJ.
Suppose thatn < 3k−2l−1, then there exists ak-island containingxand having less thank−l points to its left and less thank−lpoints to its right. This island is an isolated vertex inIJ(S′, k, l). This sole vertex is all ofIJ(S′, k, l)whenn=k(in which case the graph is connected). However, ifn > k,
there are at least two suchk-islands. ✷
The proof of Theorem 2.5 implicitly provides the following bound on the diameter ofIJ(S′, k, l).
Proposition 2.6 IfIJ(S′, k, l)is connected, then its diameter isO
n−k k−l
+O(k−l).
Proof:Suppose thatn≥3k−2l−1, as otherwise the bound trivially holds. LetIandJbe twok-islands ofS′. Note that it takes at most2(k−l)applications of Lemma 2.4 to takeIto an island whose endpoint is in the same residue class of(k−l)as the endpoint ofJ, and that containsxif and only ifJdoes. The path inIJ(S′, k, l)or inIJ(S, k, l)–depending on whetherJcontainsxor not–connecting this island to
Jhas length at most⌈n−kk−l⌉. ✷
Finally we consider the case whenl <2. As we mentioned before, ifl= 0andn≥3k−1,IJ(S, k,0) is connected and its diameter is at most3. This is the case also forIJ(S′, k,0). On the other hand, if l= 1, then the islands containingxinduce a graph isomorphic toIJ(S, k−1,0). From these observations and Lemma 2.4, we get the following result.
Proposition 2.7 Ifn≥3k,IJ(S′, k,0)andIJ(S′, k,1)are connected and of diameter at most4. ✷
2.2 Paths between projectable and non projectable islands
To finish the proof of Theorem 2.1, we prove that for any island ofP, there is a path connecting it to a projectable island. At the end of this section we present a first bound on the diameter ofIJ(P, k, l).
Recall thatP ={p0, p1, . . . , pn−1};p0is its topmost point andp1, . . . , pn−1are sorted counterclock- wise by angle aroundp0. LetAbe an island ofPsuch that|A\ {p0}| ≥2. Define theweightofAas the difference between the largest and the smallest indices of the elements ofA\ {p0}–an island of weight
k−1is always projectable. The following lemma ensures the existence of a path between any island and a projectable island, by eventually reducing the weight of any given island.
Lemma 2.8 (Shrinking Lemma). Ifn >(k−l)(k−l+ 1) +k, then every non projectablek-islandA ofP has a neighbor inIJ(P, k, l)which is either a projectable island or an island whose weight is less than that ofAby at leastk−l.
Proof:Let the elements ofAdifferent fromp0bepi1, . . . , pim. Thusmis equal tokor tok−1depending on whetherAcontainsp0or not. Consider all maximal intervals ofP\{p0}containing exactlylelements ofA. (That is maximal sets of consecutive elements ofP\p0containing exactlylelements ofA.)
We distinguish two of these intervals: the one containing the first point ofP \ {p0}and the one con- taining the last. We refer to them asend intervals, and to the rest asinterior intervals.
Note that there are at mostk−l+ 1such intervals and that every element ofP\ {p0}is in at least one of them. Sincen >(k−l)(k−l+ 1) +k, one of these intervals,I, must contain at least(k−l)points ofP\A.
Suppose thatIis an interior interval. LetJ :=A∩I, note that|J|=l(ifl= 0, setJ :=∅). IfJ is non empty letBbe the set of thek−lpoints ofI\Aclosest(ii)toConv(J). IfJis empty then letBbe anyk-island contained inI\A. ThenJ ∪Bis ak-island adjacent toAinIJ(P, k, l), and its weight is smaller than the weight ofAby at leastk−l.
Now suppose thatIis an end interval, letpSandpEbe the first and last elements inA∩I. If[pS, pE] contains at leastk−lelements ofP\A, then proceed as with interior intervals. Otherwise, there are r < k−lpoints ofP\AinI. IfIis the first interval, then letBbe thek−l−rpoints previous topS
inP \ {p0}. IfIis the last interval, then letBbe thek−l−rpoints afterpE. Note that in either case,
[pS, pE]∪Bis a projectable island adjacent toA. ✷
We are ready to finish the proof of Theorem 2.1.
Theorem 2.1 Ifn >(k−l)(k−l+ 1) +k, thenIJ(P, k, l)is connected and its diameter isO
n k−l
+
O(k−l).
Proof:LetAandBbek-islands ofP. We apply Lemma 2.8 successively to find a sequence of consecutive adjacent islandsA = A0, A1, . . . , Amand B = B0, B1, . . . , Bm′, in which each element has weight smaller than the previous by at leastk−l, and the last element is a projectable island. Since the weight of the initial terms is at mostn, these sequences have lengthO
n k−l
.
As noted before the subgraph induced by the projectable islands is isomorphic toIJ(S′, k, l). Simple arithmetic shows that ifn >(k−l)(k−l+ 1) +k, thenn >3k−2l−2. Thus this subgraph is connected and has diameter O
n−k k−l
+O(k−l) (Theorem 2.5 and Proposition 2.6). Hence the diameter of IJ(P, k, l)isO
n k−l
+O(k−l)as claimed.
✷
(ii)The distance betweenConv(J)and a pointp /∈Conv(J), is defined as the length of the shortest line segment havingpand a point ofConv(J)as endpoints.
On the Connectedness and Diameter of a Geometric Johnson Graph 27
3 Bounds
3.1 Upper bound
In this section, for the case whenl ≤k/2, we improve the upper bound on the diameter ofIJ(P, k, l) given in Theorem 2.1. We use adivide and conquerstrategy. LetAandBbe two vertices ofIJ(P, k, l).
First we find a neighbor ofAand a neighbor ofB; discarding half of the points ofPin the process. We it- erate on the new found neighbors. Just beforePhas very few points andIJ(P, k, l)may be disconnected;
we apply Theorem 2.1.
The following lemma provides the divide and conquer part of the argument. The proof uses some of the ideas of the proof of the Shrinking Lemma.
Lemma 3.1 LetAandBbe two vertices ofIJ(P, k, l). Ifn≥2((k−l)(k−l+ 1) +k), andl≤k/2, then there exists a closed halfplane,H, containing at mostn/2and at least(k−l)(k−l+ 1) +kpoints ofP. With the additional property thatAandB, each have a neighbor contained entirely inH.
Proof:We use the ham-sandwich theorem to find a lineℓso that each of the two closed halfplanes bounded ℓby contain⌈k/2⌉points ofAand⌈k/2⌉points ofB.
Without loss of generality suppose that the halfplaneH aboveℓcontains at mostn/2points ofP. If H, however, does not contain at least(k−l)(k−l+ 1) +kpoints ofP, moveℓparallel down until it does. In this caseH would contain at least as many points ofAandBas it previously did and since we are assuming thatn≥2((k−l)(k−l+ 1) +k), it still contains at mostn/2points ofP.
We will now show the existence of a neighbor of AinIJ(P, k, l)with the desired properties. The corresponding neighbor ofBcan be found in a similar way. LetP :=P ∩H and sort its elements by distance toℓ. As in the proof of Lemma 2.8 we consider maximal intervals ofP′ containing exactlyl consecutive elements ofA. There is at least one such interval, given thatH contains at leastk/2points ofAand that we are assumingl≤k/2. The rest of the proof employs the same arguments as the proof of Lemma 2.8 to find a neighbor ofAcontained in one of these intervals. ✷ Theorem 3.2 Ifn≥2((k−l)(k−l+ 1) +k) andl≤k/2, then the diameter ofIJ(P, k, l)isO(logn) + O(k−l).
Proof: Consider the following algorithm. Let A and B be twok-islands ofP. We start by setting A0 := A, B0 := B, P0 := P, n0 := n. Whileni ≥2((k−l)(k−l+ 1) +k), we apply Lemma 3.1 toPi,Ai, andBi. At each step we obtain a closed halfplaneHi containing at mostni/2 and at least (k−l)(k−l+ 1) +kpoints ofPi, with the additional property that bothAiandBihave neighborsAi+1
andBi+1contained entirely inHi. We setPi+1:=Hi∩Pi,ni+1 :=|Pi+1|, and continue the iteration.
We can do this procedure at mostO(logn)times. In the last iteration, we have a point setPmwith fewer than2((k−l)(k−l+ 1) +k)and at least(k−l)(k−l+ 1) +kelements. The islandsAmandBm
are both contained inPm, and are joined by paths of lengthO(logn)toAandBrespectively. We apply Theorem 2.1 to obtain a path of length at mostO(k−l)fromAmtoBm. Concatenating the three paths we obtain a path of lengthO(logn) +O(k−l)fromAtoBinIJ(P, k, l). ✷
3.2 Lower bound
For the lower bound we use Horton sets Horton (1983). We base our exposition on Matouˇsek (2002). Let XandY be two point sets in the plane. We say thatXishigh aboveY (and thatY isdeep belowX), if the following conditions are met:
• No line passing through a pair of points ofX∪Y is vertical.
• Each line passing through a pair of points ofXlies above all the points ofY.
• Each line passing through a pair of points ofY lies below all the points ofX.
For a setX ={x1, x2, . . . , xn}of points in the plane with no two points having the samex-coordinate and with the indices chosen so that the x-coordinate ofxi increases with i, we define the sets X0 = {x2, x4, . . .}(consisting of the points with even indices) andX1={x1, x3, . . .}(consisting of the points with odd indices). ThusX00 = {x4, x8, . . .},X01 = {x2, x6, . . .}, X10 = {x3, x7, . . .}andX11 = {x1, x5, . . .}.
Definition 3.3 A finite set of pointsH0, with no two of points having the samex-coordinate, is said to be a Horton set if|H0| ≤1, or the following conditions are met:
• BothH00andH01are Horton sets.
• H00is high aboveH01.
Horton sets of any size were shown to exist in Horton (1983). We remark that in Matouˇsek (2002), in the definition of Horton sets, the second condition is that “H00is high aboveH01orH01is high above H00”. For our purposes we need to fix one of these two options.
LetH0 :={x1, . . . , xn}be a Horton set ofnpoints. Given an islandAofH0, we define itsdepth, δ(A), to be the length of the longest strings:= 00. . .0of all zeros such thatHscontainsA. Thus for example,δ(x1) = 1,δ(x2) = 2,δ(x3) = 1,δ(x4) = 3,· · ·; refer to Figure 5. Note that the depth of an island is the depth of its shallowest point.
Lemma 3.4 Letxandybe two points ofH0, such thatxis to the left ofy, andzis a point with depth less thanδ({x, y}). Then the islandH0∩Conv({x, y, z})contains at least2δ({x,y})−δ(z)−1−1points with depth greater than that ofz, and lying in betweenxandy.
Proof: LetA := H0 ∩Conv({x, y, z}). We will proceed by induction onr = δ({x, y})−δ(z). If r = 1there is nothing to prove, since 2r−1−1 = 0. Assume then thatr > 1. Letsbe the unique string of all zeros, such thatHscontainsAbutHs0 andHs1 do not. Note thatzlies in Hs1 whilex andy both lie inHs0; actually since we are assumingr > 1, they both lie inHs00. Consider the set Hs0, since it is a Horton set,Acontains at least a point inHs01, betweenxandy. Of all such points, choosex′k to be the shallowest. The depth ofx′k is one more than that ofz. By induction, the island H0∩Conv({x, y, x′k})contains a setIof at least2r−2−1points. These points have depth greater than that ofx′k (thus contained inHs00) and lay betweenxandy. ThereforeIis contained inA. For each point inI, consider the next pointxmto its right inHs0. This point must be inHs01andδ(z)< δ(xm).
Thus we have2r−2−1additional points inA. Note that the point to the right ofxinHs0is not in the previous counting. ThereforeAhas at least2r−2−1 + 2r−2−1 + 1 = 2r−1−1points with depth greater
than that ofz, and lying betweenxandy. ✷
Lemma 3.5 IfAandBare two adjacent islands inIJ(H0, k, l)(withl≥2), then their depths differ by at mostO(log(k−l)).
Proof: Without loss of generality assume that the depth ofAis greater than the depth ofB. LetCbe the islandA∩B. Note that the depth ofCis at least the depth ofA. Ifzis the shallowest point ofB,
On the Connectedness and Diameter of a Geometric Johnson Graph 29
δ(x16) = 5 δ(x14) = 2
δ(x12) = 3 δ(x10) = 2
δ(x8) = 4 δ(x6) = 2
δ(x4) = 3 δ(x2) = 2
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
H00
H01
H011
H010
H011
H010
H011
H010
H011
H010
H001
H000
H001
H000
H001
H000
H001
H000
H0001 H0000 H0001 H0000
H00001 H00000
δ(x1) = 1 δ(x3) = 1 δ(x5) = 1 δ(x7) = 1 δ(x9) = 1 δ(x11) = 1 δ(x13) = 1 δ(x15) = 1 Fig. 5:A Horton set with16points, and the depth of its elements.
thenδ(z) = δ(B), and this point has depth less thanδ(C). Consider an edge of the convex hull ofC, whose supporting line separatesCandz. Letxandybe its endpoints. Then by Lemma 3.4, the island H0∩Conv({x, y, z})contains at least2δ({x,y})−δ(z)−1−1≥2δ(A)−δ(B)−1−1points, none of which is inC. However, since these points do lie inB, there are at mostk−lof them. Thereforeδ(A)−δ(B) isO(log(k−l))as claimed.
✷ Theorem 3.6 The diameter ofIJ(H0, k, l)forl≥2isΩ
logn−logk log(k−l)
Proof: LetAbe an island with the largest possible depth, which is⌈log2n/k⌉. LetB be an island of depth1. By Lemma 3.5 in any path joiningAandBinIJ(H0, k, l), the depth of two consecutive vertices differs byO(log(k−l)).Therefore any such path must have lengthΩ
logn−logk log(k−l)
. ✷
We point out that there was an error in the proof of Theorem 11 in the preliminary version of this paperBautista-Santiago et al. (2010); thus the bounds stated there are incorrect.
The diameter of the generalized Johnson graph can be substantially different from that of the gener- alized island Johnson graph. The diameter ofGJ(n, k, l)isO(k) whenn is large enough, while the diameter ofIJ(P, k, l)can beΩ
logn−logk log(k−l)
.
Determining upper and lower bounds for the diameter ofIJ(P, k, l)seems to be a challenging problem whenl > k/2. It might happen that there is a sharp jump in the diameter whenlrises abovek/2. We leave the closing of this gap as an open problem.
Acknowledgements
Part of the work was done during the 2nd Workshop on Discrete Geometry and its Applications, held at Instituto de Matem´aticas UNAM, Oaxaca, Mexico, September 2009. We thank all participants for useful discussions.
R.F-M and D. L. partially supported by Conacyt of Mexico, grant 153984. D. F-P. partially supported by grants 168277 (CONACYT, Mexico) and IA102513 (PAPIIT, UNAM, Mexico). J. U. partially supported by CONACyT of Mexico, grant CB-2012/178379.
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