MATEMATYKADYSKRETNAwww.ii.uj.edu.pl/preMD/

Sylwia CICHACZ and Agnieszka GORLICH

The open trails in digraphs

Preprint Nr MD 025(otrzymany dnia 4 marca 2007)

Krakow2007

Redaktorami serii preprintow Matematyka Dyskretna sa:Wit FORYS,prowadzacy seminarium S lowa, s lowa, s lowa...

w Instytucie Informatyki UJorazMariusz WOZNIAK,prowadzacy seminarium Matematyka Dyskretna - Teoria Grafow

na Wydziale Matematyki Stosowanej AGH.

The open trails in digraphs

Sylwia Cichacz∗, Agnieszka GorlichAGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland

March 4, 2007

Abstract

It has been showed in [4], when any bipartite graph Ka,b, is decom-posable into open trails of prescribed even lengths. In this article weconsider the corresponding question for directed graphs. We show thatcomplete directed graphs

←→K n and

←→K a,b are arbitrarily decomposable

into directed open trails.

1 Introduction

Consider a simple graph G whose size we denote by ‖G‖. Write V (G) for

the vertex set and E(G) for the edge set of a graph G. If G is a graph,←→G

will denote the digraph obtained from G by replacing each edge x, y ∈ E(G)by the pair of arcs −→xy and ←−yx.

Here and subsequently, a directed trail−→T of length n we identify with

any sequence (v1, v2, . . . , vn+1) of vertices of−→T such that −−−→vivi+1 are distinct

arcs of−→T for i = 1, 2, . . . , n. Notice that we do not require the vi to be

distinct. A trail−→T is closed if v1 = vn+1 and

−→T is open if v1 6= vn+1.

A sequence of positive integers τ = (t1, t2, . . . , tp) is called admissible for

a digraph←→G if it adds up to ‖←→G ‖ and for each i ∈ {1, ..., p} there exists an

open directed trail of length ti in←→G . Let τ = (t1, t2, . . . , tp) be an admissible

∗Corresponding author. E-mail: cichacz@agh.edu.pl

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sequence for←→G . If G is arc-disjointly decomposed into directed open trails−→

T 1, . . . ,−→T p of lengths t1, t2, . . . , tp respectively, then τ is called realizable in←→

G and the sequence (−→T 1, . . . ,

−→T p) is said to be a

←→G -realization of τ or a

realization of τ in←→G . If for each admissible sequence τ for

←→G there exits a←→

G -realization of τ , then digraph←→G is called arbitrarily decomposable into

open trails.

Let←→K n be the complete digraph and

←→K a,b be the complete bipartite digraph

with two sets of vertices A and B such that |A| = a and |B| = b. In our

paper we prove a necessary and sufficient condition for graphs←→K n and

←→K a,b

to be decomposable into arc-disjoint open trails of positive lengths t1, t2, ...tpfor any admissible sequence τ = (t1, t2, ...tp).

Observe that in G =←→K n or G =

←→K a,b does not exist open trail of

length ‖G‖, so we can assume that p > 1.Such problems were investigated in [4]:

Theorem 1 ([3]) For each complete bipartite graph Ka,b and for each ad-missible sequence τ = (t1, ..., tp) for Ka,b there exists a realization of τ in Ka,b

if and only if one of the following conditions holds:

10 a = 1 or

20 a and b are both even.

The first results on the topic of arbitrarily decomposition of graphs intotrails is due to Balister, who proved that if G := Kn for n odd or G := Kn−I,where I is a 1-factor in Kn, for n even, then G is arbitrarily decomposable intoclosed trails ([1]). Hornak and Wozniak ([5]) showed that complete bipartitegraphs Ka,b for a, b even are also arbitrarily decomposable into closed trails.The notion of an arbitrarily decomposable graphs into closed trails graphwere generalized to oriented graphs (see Balister [2] and Cichacz [3]). Balister

proved a necessary and sufficient condition for complete digraph←→K n:

Theorem 2 ([2]) If∑p

i=1 ti = 2(

n2

)and ti > 2 for i = 1, . . . , p, then

←→K n

can be decomposed as edge-disjoint union of directed closed trails of lengthst1, t2, . . . , tp, except in the case when n = 6 and all ti = 3.

Whereas Cichacz ([3]) showed that complete directed graphs←→K a,b are arbi-

trarily decomposable into closed directed trails.

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In this article we consider the corresponding question for open directedtrails.

2 Decomposition of bipartite complete digraphs

There is no loss of generality in assuming that a ≥ b for each complete bi-

partite digraph←→K a,b.

Theorem 3 The complete bipartite digraphs Ka,b are arbitrarily decompos-able into directed open trails.

Proof. Let A := {x1, ..., xa} and B := {v1, ..., vb}.

From now on, let us assume that G is any complete bipartite digraph←→K a,b. Let τ = (t1, ..., tp) be a sequence of positive integers such that

∑pi=1 ti =

2ab. The proof of this theorem is analogous to the proof of Theorem 1 (see[3]). We show that there exists a τ -realization in G. We consider the follow-ing cases:A. Let us suppose that ti is even for each i ∈ {1, ..., p}.Case I. Assume now that ti is not an even multiplicity of b for each i ∈{1, ..., p}. Consider a sequence

V := (v1, x1, v2, x1, v3, x1, ..., x1, vb, x1,

v1, x2, v2, x2, v3, x2, ..., x2vb, x2, ..., v1, xa, ..., xa, vb, xa, v1).

Clearly, this sequence of vertices creates an Eulerian trail in G. We show thatwe can part V into subsequences V1,...,Vp such that for each i ∈ {1, ..., p} the

set of vertices in Vi induces an open trail−→T i in G of length ti and

−→T 1, . . . ,

−→T p

are arc-disjoint subgraphs of G (see Figure 1).Let us start at the following observation: let W = (w1, ..., wk) ⊂ V be a

subsequence of consecutive elements of V such that w1 = wk = vi for somei ∈ {1, ..., b}. The set of vertices in W induces a closed trail in G of lengthm · b for some even m ≤ a.

We will define subsequences V1,...,Vp of V . Let V1 contain (t1 + 1) firstelements of V so it starts at v1 and its next elements are the consecutive

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Figure 1: Sequence V .

elements of V up to (t1 + 1)-th element. Let us denote this element by v2.Observe that it belongs to B (obviously, it is different than v1). Let V2 startat v2 and let it contains next (t2 +1) elements of sequence V . We denote thelast element of V2 by v3 so V2 = (v2, ..., v3). In the similar way we can definethe rest of subsequences V3,...,Vp. The last element of sequence Vi we willdenote by vi+1. It is easy to see that vi ∈ B for each i ∈ {2, ..., p}. Thus, asequence Vi contains the consecutive elements of V , starts at some vertex inB and finishes at the other for each i ∈ {1, ..., p}. Hence, because of aboveobservation for each i ∈ {1, ..., p} the set of vertices of Vi induces an open

trail−→T i of length ti in G. Moreover,

−→T 1, . . . ,

−→T p are arc-disjoint subgraphs

of G.Case II. Let t1 = m′

1 · b,...,tl = m′l · b for some l ∈ {1, ..., p} and for some even

integers m′1, ...,m

′l. Let mi =

m′i

2for each i ≤ l. Suppose first that l ≥ 2 and

let m := m1 + ... + ml. Then, consider a sequences:

V ′ := (xm, v1, x1, v2, x1, v3, x1, ..., x1, vb, x1,

v1, x2, v2, x2, v3, x2, ..., x2vb, x2, ..., v1, xm, ..., xm, vb, xm)

andV ′′ := (v1, xm+1, v2, xm+1, v3, xm+1, ..., xm+1, vb, xm+1,

..., v1, xa, ..., xa, vb, xa, v1).

This sequences of vertices create arc-disjoint closed trails in G. Let G′ bea subgraph of G induced by the set of vertices of V ′. Hence V ′ is an Euleriantrail for G′ which is a complete bipartite subgraph of G. Let us part V ′ intodisjoint subsequences V1 := (xm, ..., xm1), Vi := (xm1+...+mi−1

, ..., xm1+...+mi)

for i ∈ {2, ..., l−1} and Vl := (xm1+...+ml−1, ..., xm). The sequences of vertices

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Figure 2: Sequence V ′.

Figure 3: Sequence V ′′.

of Vi induce arc-disjoint open trails−→T 1,...,

−→T l of lengths t1,...,tl (see Figure 2).

Let G′′ be a graph induced by the set of vertices of V ′′. Observe that G′′ isalso a complete bipartite subgraph of G with two disjoint sets of vertices C :=A \ {x1, ..., xm} and B. The vertices of V ′′ induce an Eulerian trail for G′′ so

we can define the arc-disjoint open trails−→T l+1, . . . ,

−→T p of lengths tl+1, . . . , tp

in G′′ analogously as in case I (see Figure 3). Obviously,−→T 1, . . . ,

−→T p are

arc-disjoint open trails in G.Let us assume now that l = 1. Hence, t1 = m′ · b for some even integer

m′ and ti is not an even multiplicity of b for each i ∈ {2, ..., p}. Let m = m′2

and let us consider a sequences V ′′′ (see Figure 4) and V IV (see Figure 5)such that:

V ′′′ := (v1, x1, v2, x1, v3, x1, v4, x1, v5, . . . , xm, vb−1, xm, vb, xa, vb−1)

andV IV := (vb−1, xa, vb−2, xa, vb−3, xa, vb−4, xa, ..., v1, xa, vb

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xa−1, vb−1, xa−1, vb−2, . . . , xa−1, v1, xa−1, vb, . . .

xm+1, vb−1, xm+1, vb−2, . . . , xm+1, v1, xm+1, vb, xm, v1).

Observe that the vertices of V ′′′ induce an open trail−→T 1 of length t1. More-

Figure 4: Sequence V III .

Figure 5: Sequence V IV .

over, nearly each subsequence which contains consecutive vertices of V IV andstart and finish at the same vertex vi for any i ∈ {1, ..., b} induces a closedtrail of length k · b for some even integer k. The only exception is the set offive last vertices {v1, xm+1, vb, xm, v1} which induces a closed trail of lengthfour in G. Suppose now that there exists j ∈ {2, ..., p} such that tj 6= 4.Without loss of generality we can assume that tp 6= 4. For such admissiblesequence τ , applying analogous methods as in Case I, we can define the open

trails−→T 2, . . . ,

−→T p of length t2,...,tp in G such that

−→T 1,...,

−→T p are arc-disjoint

subgraphs of G. Assume now that ti = 4 for each i ∈ {2, ..., p}. The verticesof a sequence

V V := (v2, x1, v3, x1, v4, x1, v5, ..., xm, vb, xa, vb−1, xa, vb−2)

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induce an open trail of length t1 in G. Consider a sequence

V V I := (vb−2, xa, vb−3, xa, vb−4, xa, ...

v1, xa−1, vb, xa−1, vb−1, xa−1, vb−2, ..., xm+1, v1, xm, vb, x1, v1, x1, v2).

(see Figure 6 and 7)Let us part V V I into (p− 1) sequences, each of them containing five consec-

Figure 6: Sequence V V .

utive elements of it. Then these sets induce arc-disjoint open trails of lengthfour in G.B. Suppose now that some of elements of τ are odd. Without loss of gen-

Figure 7: Sequence V V I .

erality we can assume that t1,...,tl are odd for some l ≤ p and tl+1,...,tp areeven. Observe that l is even so there exists a positive number k such thatl = 2k. Let us define di := t2i−1 + t2i for i ∈ {1, ..., k}. Consider a sequenceτ ′ := (d1, ..., dk, t2k+1, ..., tp}. Applying the same arguments as above G is

decomposable into open trails−→D 1,...,

−→Dk,

−→T 2k+1,...,

−→T p of lengths d1,...,dk,

t2k+1,...,tp. It is easy to observe that each open trail−→D j we can part into two

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arc-disjoint open trails T2j−1, T2j of lengths t2j−1, t2j. Hence,−→T 1, . . . ,

−→T p is

a G-realization of τ and the proof is finished.

3 Decomposition of complete digraphs

Let τ = (t1, . . . , tp) is an admissible sequence for G. We can assume thatt1 ≤ . . . ≤ tp. Let si = t1 + . . . + ti. We shall write ((t1)

r1 , ..., (tl)rl) for the

sequence (t1, ..., t1︸ ︷︷ ︸r1

, ..., tl, ..., tl︸ ︷︷ ︸rl

).

Using Theorem 3 we prove the analogously result for complete digraphs.We start our analysis by dealing with n ≤ 4.

Lemma 4 If τ = (t1, ..., tp) is an admissible sequence for G =←→K n and

n ≤ 4, then τ is realizable in G, unless τ = (2, 2, 2) (hence G =←→K 3).

Proof.←→K 2 is trivially arbitrarily decomposable into open trails. One may

check that every admissible sequence τ for the digraph←→K 3, except the se-

quence τ = (23) is←→K 3-realizable.

The sequences (26), (43) and (62) are←→K 4-realizable, see Figure 8. It is easy

Figure 8:←→K 4-realizations of three sequences

to check that every admissible sequence τ such that tp = 6 is realizable in←→K 4 (see realization of (62) on Figure 8). The same is with sequences suchthat tp = 2 (see realization of (26) on Figure 8)

Let τ = (t1, . . . , tp) be an admissible for←→K 4 but different than described

above. Because an admissible sequence τ for←→K 4 is non-decreasing, then

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Figure 9: The graph←→K 4 as

←→K 1,3 ∪←→K 3

tp ≥ 3. Notice that we can consider digraf←→K 4 as an arc-disjoint union of

digraphs←→K 1,3 and

←→K 3 (see Figure 9).

Let us assume first that there exists i ∈ {1, ..., p} such that si−1 <

‖←→K 1,3‖ = 6 and si > 7 (notice that i ≥ 2). Let t′i = 6 − si−1, t′′i = ti − t′i.By Theorem 3 we can find a realization (

−→T 1, . . . ,

−→T i−1,

−→T ′

i) of the sequence

τ ′ = (t1, . . . , ti−1, t′i) in

←→K 1,3. If t′′i = 6 then let

−→T i =

−→T ′

i ∪ T←→K 3

, where T←→K 3

is an Eulerian trail in←→K 3. If t′′i < 6, then we can find realization of sequence

τ ′′ = (t′′i , ti+1, . . . , tp) (notice that τ ′′ 6= (23)) in←→K 3. We may write the open

directed trail−→T ′

i as (v1, . . . , vt′i+1) and−→T ′′

i as (w1, . . . , wt′′i +1). Observe that

if t′i is odd, then v1 ∈ V (←→K 3) or vt′i+1 ∈ V (

←→K 3), whereas if t′i is even then

v1, vt′i+1 ∈ V (←→K 3). If v1 ∈ V (

←→K 3), then we may have chosen the realization

of τ ′ in such a way that v1 = wt′′i +1 and vt′i+1 6= w1. In such a case we define−→T i = (w1, . . . , wt′′i +1, v2, . . . , vt′i+1) and

−→T i is an open directed trail of lengths

ti. Hence, let v1 /∈ V (←→K 3). It implies that vt′i+1 ∈ V (

←→K 3). It is obvious then

we can assume that vt′i+1 = w1 and let−→T i := (v1, . . . , vt′i+1, w2, . . . , wt′′i +1)).

The sequence (−→T 1, . . . ,

−→T p) is an

←→K 4-realization of τ .

Assume now that si = ‖←→K 1,3‖ = 6. Because tp < 6 so the sequences

τ1 := (t1, . . . , ti) and τ2 := (ti+1, . . . , tp) are realizable in←→K 1,3 and

←→K 3

resp.

Theorem 5 If∑p

i=1 ti = ‖←→K n‖ and ti > 1 for i = 1, . . . , p, then←→K n

can be decomposed as edge-disjoint union of directed open trails of lengthst1, t2, . . . , tp, except in the case when n = 3 and all ti = 2.

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Proof. Let τ = (t1, . . . , tp) is an admissible sequence for G :=←→K n (p > 1).

We will argue by induction on n. The basic idea of the proof is to consider←→K n as an arc-disjoint union of

←→K 1,n−1 and

←→K n−1, each of which have sizes

2(n−1) and (n−1)(n−2) resp. Let A and B be the partition sets of←→K 1,n−1

such that |A| = 1 and |B| = n− 1. We verified the cases for n ≤ 4 using theLemma 4, hence let us assume n ≥ 5.

Case 1: For some i ∈ {1, ..., p}, si−1 < 2(n−1) and si > 2(n−1) (s0 = 0). Letτ1 = (t1, . . . , ti−1, t

′i) and τ2 = (t′′i , ti+1, . . . , tp), where t′i = 2(n− 1)− si−1 > 1

and t′′i = ti−t′i > 1. Assume first that i = 1, then t′i = 2(n−1). By induction

we can find a realization (−→T ′′

1,−→T 2 . . . ,

−→T p) in

←→K n−1. Let

−→T 1 :=

−→T ′′

1∪←→K 1,n−1

and we obtain a←→K n-realization of τ .

If i > 1 then by Theorem 3 we can find a realization (−→T 1, . . . ,

−→T i−1,

−→T ′

i) of τ ′

in←→K 1,n−1 and (

−→T ′′

i ,−→T i+1 . . . ,

−→T p) is

←→K n−1-realizations of τ ′′ by induction.

We define the open directed trail−→T ′

i as (v1, . . . , vt′i+1) and−→T ′′

i as (w1, . . . ,

wt′′i +1). Now we use analogous method as in the proof for←→K 4 (see Lemma 4).

Note again that if t′i is odd, then v1 ∈ B or vt′i+1 ∈ B, whereas if t′i is even,then v1, vt′i+1 ∈ B. Assume first that v1 ∈ B. Because n− 1 > 3, so we mayhave chosen the realization of τ ′ in such a way that v1 = wt′′i +1 and vt′i+1 6= w1.

In such a case, if we denote−→T i = (w1, . . . , wt′′i +1, v2, . . . , vt′i+1), then

−→T i is

an open directed trails of lengths ti and the sequence (−→T 1, . . . ,

−→T p) is an←→

K n-realization of τ . If v1 /∈ B then we can number the vertices in the set Bin such a way that vt′i+1 = w1 and let

−→T i := (v1, . . . , vt′i+1, w2, . . . , wt′′i +1).

Case 2: For some i ∈ {1, ..., p}, si = 2(n− 1).Assume first that i > 1. If p > i + 1, then by Theorem 3 we can find a

realization (−→T 1, . . . ,

−→T i) of τ ′ in

←→K 1,n−1 and (

−→T i+1 . . . ,

−→T p) is

←→K n−1-reali-

zations of τ ′′ by induction. When p = i + 1 then because n ≥ 5, we have

tp > 2(n − 1) = ‖←→K 1,n−1‖ we proceed the same way as in Case 1 (splittingtp into two parts this time).Suppose now that i = 1. It implies t1 = 2(n − 1). If additionally t1 < tp,we have si − t1 + tp < 2(n − 1) and we again proceed the same way asin Case 1 (splitting tp into two parts). Hence, we are left with the caset1 = tp = 2n− 2 ≥ 8.

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Figure 10: The realization (−→T ′

1,−→T 2) of the sequence τ ′ in the graph Gn−2

Let a graph Gn−2 be defined as an arc-disjoint union of digraphs←→K 2,n−2

(with partitions sets A′ = {x, y} and B′ = {b1, . . . , bn−2}) and←→K 2 (V (

←→K 2) =

{x, y}), see Figure 10. We consider now←→K n as an arc-disjoint union of

Gn−2 and←→K n−2, each of which have sizes 4n − 6 and (n − 2)(n − 3) resp.

We can find a realization (−→T ′

1,−→T 2) of the sequence τ ′ = (t′1, t2) = (2n −

4, 2n − 2) in Gn−2, such that−→T ′

1 = (b1, x, b2, x, b3, x, . . . , bn−2, y, bn−2) and−→T 2 = (bn−2, x, y, b1, y, b2, y . . . , bn−3, y, x, b1) (see Figure 10). By induction

there exists a realization (−→T ′′

1,−→T 3, . . . ,

−→T p) of sequence τ ′′ = (t′′1, t3, . . . , tp) =

(2, (2n− 2)p−2) in←→K n−2 (notice that τ ′′ 6= (23)). Without loss of generality

(because n− 2 ≥ 3) we can write−→T ′′

1 as a vertex sequence (bn−2, b1, b2). Let

now−→T 1 = (b1, x, b2, x, b3, x, . . . , bn−2, y, bn−2, b1, b2) and we obtain the

←→K n-

realization (−→T 1, . . . ,

−→T p) of τ .

References

[1] Balister, P.N., Packing Circuits into KN , Combin. Probab. Comput. 10(2001) 463–499.

[2] Balister, P.N., Packing digraphs with directed closed trails, Combin.Probab. Comput. 12 (2003) 1–15.

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[3] Cichacz, S., Decomposition of complete bipartite digraphs and even com-plete bipartite multigraphs into closed trails, Preprint MD 016 (2005),http://www.ii.uj.edu.pl/preMD/.

[4] Cichacz, S., and A. Gorlich Decomposition of complete bi-partite graphs into open trails, Preprint MD 022 (2006),http://www.ii.uj.edu.pl/preMD/.

[5] Hornak, M., and M. Wozniak, Decomposition of complete bipartite evengraphs into closed trails, Czechoslovak Mathematical Journal, 128 (2003)127–134.

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