4-cycles in mixing digraphs

Omid Amini 1,2

I3S (CNRS-UNSA) INRIASophia Antipolis, France

Simon Griffiths 1,2

DPMMSCambridge University, England

Florian Huc 1,2

I3S (CNRS-UNSA) INRIASophia Antipolis, France

Abstract

It is known that every simple graph with n3/2 edges contains a 4-cycle. A similarstatement for digraphs is not possible since no condition on the number of arcscan guarantee an (oriented) 4-cycle. We find a condition which does guarantee thepresence of a 4-cycle and our result is tight. Our condition, which we call f -mixing,can be seen as a quasirandomness condition on the orientation of the digraph. Wealso investigate the notion of mixing for regular and almost regular digraphs. Inparticular we determine how mixing a random orientation of a random graph is.

Keywords: oriented 4-cycles, digraphs, pseudo random digraphs.

1 This research was supported by the European project AEOLUS, the french region PACA,ecole Polytechnique and EPSRC.2 Email: omid.amini@sophia.fr,sg332@cam.ac.uk,florian.huc@sophia.fr

Electronic Notes in Discrete Mathematics 30 (2008) 63–68

1571-0653/$ – see front matter © 2008 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2008.01.012

1 IntroductionThroughout this paper we use the notation G for an undirected graph andD for a digraph, V for a vertex set and E for an edge (or arc) set. Unlessotherwise stated all digraphs are without loops, multiple arcs or 2-cycles. Wewrite e(D), or simply e, for the number of arcs in a digraph D. We usethe notation xy or yx to represent an edge {x, y}. An arc (x, y) will bedenoted −→xy. For two not necessarily disjoint subsets A, B ⊂ V , we denote byE(A, B) = {−→uv ∈ E : u ∈ A, v ∈ B} the set of arcs from A to B, and bye(A, B) = |E(A, B)| its cardinality. The out-neighbours (resp. in-neighbours)of u is the set Γ+(u) = {v : −→uv ∈ E} (resp. Γ−(u) = {v : −→vu ∈ E}). The out-degree (resp. in-degree) of u is d+(u) = |Γ+(u)| (resp. d−(u)). We shall alsomention the out-out-neighbours Γ++(u) = {w : ∃v ∈ V such that−→uv,−→vw ∈ E}and the in-in-neighbours Γ−−(u).In a simple graph G (resp. a digraph D) a k-cycle consists of k distinct verticesv0, . . . , vk−1 such that for all 0 ≤ i ≤ k − 1, vivi+1 modulo k (resp −−−→vivi+1) ∈ E.

Finally, for a more complete version, please refer to [1].

1.1 Our problemIt is known that every simple graph with n3/2 edges contains a 4-cycle. Specif-ically, writing ex(n, C4) for the maximum number of edges a graph G on nvertices can have without containing a 4-cycle, it was shown by Erdos, Renyiand Sos [2] that ex(n, C4) = (1

2+ o(1))n3/2, (see also [3] which shows that the

example of [2] is best possible).A similar result for digraphs is not possible, consider the digraph D on

vertex set V = {1, ..., n}, with an arc−→ij whenever i < j. D has

(n2

)arcs

(the most possible) but contains no (oriented) 4-cycle. So no condition onthe number of arcs can guarantee the existence of an 4-cycle. However, inthis example D has an extreme ‘bias’ in its orientation: we can find subsetsA, B ⊂ V with e(A, B) = n2/4 but e(B, A) = 0. Our main result will be thatonly strongly biased digraphs can avoid containing 4-cycles.

Definition 1.1 Given a digraph D and f ∈ R, we say D is f -mixing if forevery (not-necessarily disjoint) pair of subsets A, B ⊂ V with e(A, B) ≥ f ,we have e(B, A) > e(A, B)/2.

In words, a digraph D is f -mixing if whenever there are many arcs fromA to B (ie. at least f arcs) then we have more than half as many back. Thisgives us a large range of mixing conditions getting stronger as f decreases.We can now formulate the main theorem of this paper:

Theorem 1.2 i There exists ε > 0, such that every (εe2/n2)-mixing digraphD contains a 4-cycle.

ii In fact D has at least ce4/n4 of them (for a constant c > 0).

O. Amini et al. / Electronic Notes in Discrete Mathematics 30 (2008) 63–6864

Remark 1.3 i The number of 4-cycles in a typical simple graph is of theorder e4/n4. Hence the number of (oriented) 4-cycles given to us by theabove theorem is (up to multiplication by constant) the largest we couldpossibly hope for. In fact any (εe2/n2)-mixing digraph contains thecorrect order of any orientation of a 4-cycle.

ii In Section 3, we give examples of digraphs which are (Ke2/n2)-mixing(where K is a large constant) and do not contain 4-cycles. This meansthat the mixing condition cannot be weakened, so Theorem 1.2 is best pos-sible.

iii In the case where e is of order n2, it is not too difficult to deduce Theorem1.2 using Szemeredi’s regularity Lemma (cf [4] for a survey).

Given a result like Theorem 1.2 it is important to ask: Do there existdigraphs which are (εe2/n2)-mixing? Lemma 1.4, which is obtained by asimple application of Chernoff’s inequality, answers this question:

Lemma 1.4 There exists K such that for any simple graph G on n vertices,the digraph D obtained by orienting the edges of G at random, is Kn-mixingwith positive probability (always) and with high probability (as n tends to infin-ity). In particular, if e is much larger than n3/2 then Kn is less than (εe2/n2)and so D is (εe2/n2)-mixing with positive probability and with high probability.

In Section 2, we sketch the proof of the first part of Theorem 1.2, thecomplete proof can be found in [1] as well as all the omitted proofs. In Sec-tion 3, we show that the condition (εe2/n2)-mixing can not be weakened to(Ke2/n2)-mixing, where K is a large constant. In Section 4, we consider thequestion of ‘how mixing’ a digraph can be. In particular, we find a constantc > 0 such that randomly oriented random graphs are (with high probability)not cn-mixing. This provides a converse to Lemma 1.4.

2 Proof of Theorem 1.2(i)Given a digraph D on V = {1, ..., n}, it simplifies the proof to consider(D, X, Y ), the double cover of D. (D, X, Y ) is defined on vertex set X ∪ Ywhere X = {x1, ..., xn} and Y = {y1, ..., yn}, and has arc set E(D, X, Y ) =

{−−→xiyj,−−→yixj :

−→ij ∈ E(D)}. If D is f -mixing, then (D, X, Y ) has Property 1:

Property 1 For any pair A ⊂ X and B ⊂ Y (or A ⊂ Y and B ⊂ X) suchthat e(A, B) ≥ f , we have e(B, A) > e(A, B)/2.

We take ε = 1/32, if D is εe2/n2-mixing then (D, X, Y ) has Property 1with f = e2/32n2, we show that this implies the presence of a four cycle in(D, X, Y ), which in turn implies the presence of a four cycle in D. We beginby defining for each vertex x ∈ X the quantity ex = e(Γ+(x), Γ++(x)).

O. Amini et al. / Electronic Notes in Discrete Mathematics 30 (2008) 63–68 65

Lemma 2.1 With (D, X, Y ) as above, we have:∑

x∈X ex ≥ e2/8n

We now give a compact version of our proof of Theorem 1.2(i), for thecomplete version cf [1]. We focus on the set of vertices W = {x ∈ X : ex ≥e2/16n2}. In doing so we keep most of the sum

∑x∈X ex:∑

x∈W

ex ≥ e2/16n.

Now for each x ∈ W we have e(Γ+(x), Γ++(x)) ≥ εe2/n2. Property1 implies e(Γ++(x), Γ+(x)) ≥ ex/2. In other words, writing d++(x, u) for|Γ+(x) ∩ Γ+(u)|, we have for each x ∈ W that,

∑u∈Γ++(x) d++(x, u) ≥ ex/2.

Hence:∑u

∑x∈Γ−−(u)

d++(x, u) =∑x

∑u∈Γ++(x)

d++(x, u) ≥ 12

∑x∈W

ex ≥ e2

32n

So there exists u with∑

x∈Γ−−(u) d++(x, u) ≥ e2/32n2, ie. e(Γ−−(u), Γ+(u)) ≥e2/32n2. By Property 1, we have: e(Γ+(u), Γ−−(u)) ≥ e2/64n2. Each arc fromΓ+(u) to Γ−−(u) gives us a 4-cycle so we are done because e2/64n2 > 0 (asthe empty graph is not εe2/n2-mixing). The proof works for ε = 1

32.

3 (Ke2/n2)-mixing without a 4-cycleWe have proved that every (εe2/n2)-mixing digraph contains a 4-cycle. Itis natural to ask whether this result would fail if ε was replaced by a largeconstant K. In this Section, we give examples of digraphs which are (Ke2/n2)-mixing but do not contain any oriented 4-cycle.

Using Erdos-Renyi graphs, we can find for all m, a graph G on m verticeswith at least 1

20m3/2 edges which does not contain a 4-cycle. By Lemma 1.4, a

random orientation of G is Lm-mixing with positive probability, where L is aconstant. Taking D′ as a Lm-mixing orientation of G gives our first example,for D contains no (oriented) 4-cycle and is (Ke2/m2)-mixing, for K = 400L.

Theorem 1.2 holds for all pairs n, e, the examples above all have e =Θ(n3/2). Are there examples for other pairs n, e? In fact taking a strongervariant of the above example, and then taking a blow up of it in which eachvertex is replaced by l vertices and each arc by the corresponding l2 arcs,we obtain a digraph D with |V (D)| = ml and e(D) = e(D′)l2, which isK ′e(D)2/|V (D)|2-mixing (for some K ′, proof sketched below) which does notcontain 4-cycles. Hence, we obtain a wide class of examples, one for each pairm, l. Now for any pair n, e with e > n3/2 a suitable choice of m, l yields adigraph D with about n vertices and about e edges which is K ′e2/n2-mixingand contains no 4-cycles.

D is K ′e(D)2/|V (D)|2-mixing for, if there were sets A, B ⊂ V (D) withe(A, B) ≥ K ′e2/n2 and e(B, A) ≤ e(B, A)/2, then one can find A′, B′ ⊂ V (D)

O. Amini et al. / Electronic Notes in Discrete Mathematics 30 (2008) 63–6866

each a union of cells of the blow up which also display a strong bias, thiscontradicts the mixing property of D′, and thus no such pair A, B can exist,ie. D is K ′e2/n2-mixing. See [1].

4 How mixing can digraphs be?Lemma 1.4 says that, if K is a large constant, randomly oriented digraphs areKn-mixing with high probability. The presence of non necessarily disjoint setsof vertices A and B with arcs from A to B and none from B to A (e(A, B) > 0and e(B, A) = 0) is a natural obstruction for a digraph to be highly mixing.It is why we ask the following question:

Given a digraph D, what is maxABD = max{A,B:e(B,A)=0}(e(A, B))? In fact

this question can be asked for any digraph, with or without 2-cycles. Let nowstate some complexity results about it:

Theorem 4.1 Given a digraph potentially with 2-cycles, computing maxABD

is NP-hard and hard to approximate within a factor n1−ε for some ε > 0.

In a digraph with multiple arcs, maxABD can be equal to zero, but this is

strongly due to the presence of 2-cycles. What happens when the digraph hasno 2-cycles? We prove that for almost d-regular digraphs (digraphs such thatfor all vertices v ∈ V we have d ≤ d−(v), d+(v) ≤ 2d), maxAB

D is at least linearin n. This means that almost regular graphs cannot be εn-mixing for small ε.To prove it, we construct the sets A and B using the following algorithm:

Algorithm 1 1: Let v ∈ V , set A = {v} and B = Γ+(v).2: For all u ∈ V \ A evaluate the function:

f(u) = e({u}, A) + e(A, {u, })∑

v∈A

(|Γ+(v)∩ Γ−(u)|+ |Γ+(u)∩ Γ−(v)|

).

3: while there is a vertex u with f(u) < d+(u) do4: Add to A the vertex v which maximizes d+(v)− f(v).

5: Update B: B =(B ∪Γ+(v)

)\

(A∪

(B ∩Γ−(v)

)∪

(Γ+(v)∩Γ−(A)

)).

6: Update f(u) for all u ∈ V \ A.7: end while8: return A and B.

Using this algorithm on almost d-regular digraphs, one can prove:

Lemma 4.2 Let D be an almost d-regular digraphs, there exist subsets A, B ⊂V with e(A, B) ≥ n

16and e(B, A) = 0. So that D is emphatically not ( n

16)-

mixing.

As a random orientation of a random graph gives whp an almost regulardigraph, we may deduce:

O. Amini et al. / Electronic Notes in Discrete Mathematics 30 (2008) 63–68 67

Corollary 4.3 Consider the digraph D obtained by randomly orienting theedges of the random graph G(n, p). If p = ω(log n/n) then whp there existsubsets (A, B) ⊂ V (G)2 with e(A, B) ≥ n

16and e(B, A) = 0. So that with high

probability D is emphatically not ( n16

)-mixing for large n.

5 ConclusionWe have introduced a new notion: the notion of mixing digraphs. This notioncan be used to give a tight condition for guaranteeing the presence of a 4-cycle in the digraph. For a fixed digraph D′ one can ask whether there existsa mixing condition which guarantees the presence of a copy of D′. So far wehave only partial results. We have also investigated how mixing digraphs canbe. In particular, almost d-regular digraphs are not ( n

16)-mixing and whp a

random orientation of a random graph is Kn-mixing but not εn-mixing whereK > ε > 0 are constants. In proving these results we have shown that itis often possible to find subsets A, B with many edges from A to B, whilethere are none from B to A. Many interesting questions remain such as, ismaxAB

D NP-hard in digraphs without 2-cycles, can we find algorithms to betterapproximate maxAB

D .

References[1] www-sop.inria.fr/sloop/personnel/Florian.Huc/Publications/4cycles.pdf

[2] P. Erdos and A. Renyi and V.T. Sos, On a problem of graph theory, Studia Sci.Math. Hungar. 1, 215-235, 1966

[3] Z. Furedi, Graphs without Quadrilaterals, Journal of Comb. Theory, Series B34, 187-190, 1983

[4] J. Komlos, A. Shokoufandeh, M. Simonovits, E. Szemeredi, The regularitylemma and its applications in graph theory, Theoretical aspects of computerscience (Teheran 2000), 84–112, Springer

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