Directed cycles with chords

Directed Cycles withChords

Daniel A. MarcusCALIFORNIA STATE POLYTECHNIC UNIVERSITY

POMONA, CA

Received May 30, 1997; revised July 6, 1998

Abstract: Using a variation of Thomassen’s admissible triples technique,we give an alternative proof that every strongly 2-arc-connected directed graphwith two or more vertices contains a directed cycle that has at least two chords,while at the same time establishing a more general result. c© 1999 John Wiley & Sons, Inc.

J Graph Theory 31: 17–28, 1999

Keywords: graph, directed graph, cycle, chord, spanning subgraph

1. INTRODUCTION

In [1], this author considered the problem of determining an upper bound for thenumber of arcs needed for a strongly connected spanning directed subgraph in anystrongly 2-arc-connected directed graph havingm arcs andn vertices. It was shownthere that the best possible bounds of the form am+ b(n− 1) could be establishedas a consequence of a conjecture concerning directed cycles having at least twochords. After many years, that conjecture was finally settled in the affirmative byThomassen [2], using an innovative technique in which the strongly 2-arc-connectedcondition is partially relaxed. In this article, we develop a variation of Thomassen’smethod that facilitates a reorganization and substantial simplification of the proofand allows the result to be generalized in two directions. In particular, we obtainresults that apply to strongly k-arc-connected directed graphs, for all k ≥ 2, andat the same time we replace strong k-arc-connectivity with a simpler condition onindegrees and outdegrees of vertices.

c© 1999 John Wiley & Sons, Inc. CCC 0364-9024/99/010017-12

18 JOURNAL OF GRAPH THEORY

2. DIRECTED CYCLES WITH CHORDS

Definitions. A digraph is a graph with directed edges, or arcs. Each arc will beassumed to have two distinct endpoints and multiple arcs having the same endpointsare allowed. A vertex p dominates another vertex q if there exists an arc from p toq. The term path will refer to a simple directed path in a digraph. A digraph D isstrongly connected if any vertex is reachable from any other vertex by some path inD. Equivalently, every nonempty proper vertex set in D has at least one incomingarc and at least one outgoing arc. The term cycle will refer to a simple directedcycle in a digraph. Also, for technical reasons, we will consider a single vertex tobe a cycle. A chord of a subdigraph is an arc both of whose endpoints are in thesubdigraph, but which is not itself an arc of the subdigraph.

To simplify notation, at times no distinction will be made between a digraph andits vertex set, where there is no possibility of confusion.

Main Theorem. Let D be a strongly connected digraph containing at least twovertices and assume that each vertex has at least k incoming arcs and at least koutgoing arcs, for some integer k ≥ 2. Then D contains a cycle that has at least kchords.

In particular, this result applies to any strongly k-arc-connected digraph withk ≥ 2 and at least two vertices. Using the argument employed in [1] and indicatedat the beginning of [2], we obtain the following.

Corollary. For k ≥ 2, every strongly k-arc-connected digraph having m arcsand n ≥ 2 vertices contains a strongly connected spanning subdigraph that has atmost (m+ k(n− 1))/(k + 1) arcs.

Definitions. Let S and T be two disjoint sets of vertices in a digraphD such that Sand D− S are nonempty. We call (D,S, T ) a positive Thomassen triple of degreek, if:

(1) every nonempty set of vertices in D − S has at least one outgoing arc andevery vertex in D − S has at least k outgoing arcs; and

(2) every nonempty set of vertices in D− (S ∪ T ) has at least one incoming arcand every vertex in D − (S ∪ T ) has at least k incoming arcs.

Interchanging the terms ‘‘incoming’’ and ‘‘outgoing’’ results in the definition ofa negative Thomassen triple of degree k.

Notation. In a digraph D, let p be a vertex and X a set of vertices not containingp. We denote by R+(p,X) the set of all vertices in the subdigraph D−X that arereachable from p inD−X , and byR−(p,X) the set of all vertices inD−X fromwhich p is reachable in D −X .

Remark. Notice that any arc that leaves R+(p,X) must necessarily terminate inX , while any arc that enters R−(p,X) must originate in X .

Lemma 1. Let (D,S, T ) be a positive Thomassen triple of degree k ≥ 1. Let Xbe a vertex set containing S, and let p be a vertex not in X . Set A = R+(p,X)and B = R−(p,X); then

DIRECTED CYCLES WITH CHORDS 19

(a) Every vertex in D −A is reachable from X ∪ T in D −A; and(b) Every vertex in D −B can reach to X in D −B.

Proof. (a) Let q be a vertex in D−A and let B′ denote the set R−(q, A). Weclaim that B′ contains some vertex in X ∪ T . If not, then B′ would be containedin D − (S ∪ T ) and would, therefore, have at least one incoming arc in D. By theremark above, this arc must originate inA. But applying the remark again, we findthat this same arc terminates in X , contradicting the assumption that B′ is disjointfrom X .

(b) Let q be a vertex in D − B and let A′ denote the set R+(q,B). We claimthat A′ contains some vertex in X . If not, then A′ would be contained in D − Sand would, therefore, have at least one outgoing arc in D. By the remark above,this arc must terminate inB. But applying the remark again, we find that this samearc originates in X , contradicting the assumption that A′ is disjoint from X .

Lemma 2. Let (D,S, T ) be a positive Thomassen triple of degree k ≥ 1 and letS′ be a nonempty set of vertices in D. Let X be the union of S and S′, and let p beany vertex in D −X.

(a) Let A = R+(p,X) and denote by DA the subdigraph of D induced by S′and A, and by TA the set

(A∩ T )∪ {terminal vertices of arcs that enter A and originate outside DA}.If no arc goes from A to S − S′, then (DA, S

′, TA) is a positive Thomassentriple of degree k.

(b) Let B = R−(p,X) and denote by DB the subdigraph of D induced by S′and B, and by TB the set

{initial vertices of arcs that leave B and terminate outside DB}.If B ∩ T is empty and no arc goes from S − S′ to B, then (DB, S

′, TB) is anegative Thomassen triple of degree k.

Proof. (a) We note first that any arc in D that originates in A must terminatein either A or X , and since no arc goes from A to S − S′, it follows that any sucharc terminates in either A or S′ and consequently exists in DA. Now let W be anonempty set of vertices in DA− S′ = A. Then W is in D− S and so has at leastone outgoing arc in D. By the foregoing observation, this arc exists in DA. Thesame applies to all outgoing arcs at vertices ofA, so (DA, S

′, TA) satisfies condition(1) for a positive Thomassen triple of degree k. To establish condition (2), observethat the definition of TA implies that any arc in D that terminates in A− TA mustexist in DA. Now let W be a nonempty set of vertices in DA − (S′ ∪ TA) =A − TA. Then W is in D − (S ∪ T ) and so has at least one incoming arc in D.Again, this arc must exist in DA and again the same applies to all incoming arcs atvertices of A− TA.

(b) We note first that any arc inD that terminates inB must originate in eitherBorX , and since no arc goes from S−S′ toB, it follows that any such arc originates


in either B or S′ and consequently exists in DB . Now let W be a nonempty setof vertices in DB − S′ = B. Then W is in D − (S ∪ T ) and so has at least oneincoming arc in D. By the foregoing observation, this arc exists in DB . The sameapplies to all incoming arcs at vertices ofB, so (DB, S

′, TB) satisfies condition (1)for a negative Thomassen triple of degree k. To establish condition (2), observe thatthe definition of TB implies that any arc in D that originates in B − TB must existin DB . Now let W be a nonempty set of vertices in DB − (S′ ∪ TB) = B − TB .ThenW is inD−S and so has at least one outgoing arc inD. Again, this arc mustexist in DB and again the same applies to all outgoing arcs at vertices of B − TB .

Definitions. LetS be a set of vertices in a digraphD and let v be a vertex inD−S.A (v, S)-path is a path that starts at v, contains no vertices of S, and whose finalvertex dominates some vertex in S. A (v, S)-path of type k is a (v, S)-pathQ suchthat at least k arcs start at the final vertex of Q and terminate in Q ∪ S. Reversingall directions results in the definition of an (S, v)-path and an (S, v)-path of type k.

Lemma 3. Let C be a cycle in a digraph D and let v be a vertex in D − C. IfD contains a (C, v)-path of type h and a (v, C)-path of type k intersecting onlyat v, then D contains a cycle that has h + k − 2 chords, each having at most oneendpoint in C.

We now state three related theorems on Thomassen triples. The Main Theoremwill follow as an immediate consequence of the last of these. While the theoremsare stated here for positive triples, each has a corresponding dual form for negativetriples.

Theorem 1. Let (D,C, T ) be a positive Thomassen triple of degree k ≥ 2,where C is a cycle in D. Let v be a vertex in D − C and let Q be a (v, C)-pathwhose final vertex z dominates some vertex u in D − Q having the property thatR+(u,Q) contains no vertex of C. Then at least one of the following conditions issatisfied:

(1) D contains a cycle having k chords, each of which has at most one endpointin C;

(2) D contains a (t, C)-path having k chords, for some vertex t in T ;(3) D contains a (v, C)-path having k chords.

Theorem 2. Let (D,C, T ) be a positive Thomassen triple of degree k ≥ 2,whereC is a cycle in D. Then at least one of the following conditions is satisfied:


(2) D contains a (t, C)-path having k chords, for some vertex t in T ;(3) For each vertex v inD−C,D contains either a (v, C)-path having k chords

or a (v, C)-path of type k.

Theorem 3. Let (D,C, T ) be a positive Thomassen triple of degree k ≥ 2,whereC is a cycle in D. Then at least one of the following conditions is satisfied:



(2) D contains a (t, C)-path having k chords, for some vertex t in T ;(3) For each vertex v inD−C that is dominated by some vertex ofC, there exists

a (t, C)-path of type k that contains v, for some vertex t in T.

Proof of the Main Theorem, using Theorem 3. Let C consist of a single vertexin D. Then (D,C,∅) is a positive Thomassen triple of degree k. Conditions (2)and (3) of Theorem 3 are impossible, so (1) must hold. (Condition (3) does nothold by vacuity, since at least one outgoing arc from C must exist.)

We note that the same argument establishes the following stronger version ofthe main theorem: Let D be a strongly connected digraph and let C be a cycle inD such that D − C is nonempty. Suppose that each vertex in D − C has at leastk incoming arcs and at least k outgoing arcs, for some integer k ≥ 2. Then Dcontains a cycle that has k chords, each having at most one endpoint in C.

The proofs of Theorems 1–3 will be carried out by simultaneous induction onthe number n of vertices in D − C, beginning with the case n = 1 of Theorem 2.

Proof of Theorem 2 when n = 1n = 1n = 1. Let v be the vertex in D − C. At least k arcsgo from v to C. Then the path consisting of the single vertex v is a (v, C)-path oftype k, satisfying condition (3) of Theorem 2.

Proof of Theorem 3 when n = 1n = 1n = 1. Let v be the vertex in D − C. As above, thepath consisting of the single vertex v is a (v, C)-path of type k. If v is in T , thencondition (3) of Theorem 3 is satisfied. If v is not in T , then T is empty and at leastk arcs go from C to v. Therefore, {v} is also a (C, v)-path of type k. ApplyingLemma 3 and noting that 2k − 2 ≥ k, we conclude that condition (1) is satisfied.

The first case of Theorem 1 occurs when n = 2. This case is a consequence ofthe above, along with the following.

2.1. Induction Step for Theorem 1, Using the Inductive Hypotheses ofTheorems 2 and 3

Add an arc zv to D, extending the path Q to a cycle C ′ (unless z = v, in whichcaseC ′ = C = {v}), and letD∗ denote the augmented digraph. We apply Lemma2 to the triple (D∗, C, T ) with S = C, S′ = C ′, and p = u. The assumptionthat R+(u,Q) contains no vertex of C implies that R+(u,Q) = R+(u,X), whereX is the union of the vertex sets of C and C ′. Moreover, the same assumptionimplies that no arc goes from A = R+(u,X) to C. Therefore, (a) of Lemma 2applies, establishing that (DA, C

′, TA) is a positive Thomassen triple of degree k.DA − C ′ = A contains fewer vertices than D − C, so the inductive hypothesisof Theorem 3 applies to (DA, C

′, TA). We consider the three conditions in theconclusion of Theorem 3 as applied to this triple.

If (1) holds, thenDA contains a cycle that has k chords in the augmented digraphD∗. If such a cycle does not include arc zv, then it exists inD; and if it does include


zv, then DA contains a path from v to z that has k chords in D. In either case,condition (1) or (3) of Theorem 1 is satisfied for the triple (D,C, T ). (Notice thatthere is no danger that zv is a chord of the cycle obtained from Theorem 3, sincesuch a chord can have at most one endpoint in C ′.)

If (2) holds, thenA contains a(t, C ′)-path that has k chords and starts at a vertext in TA. If t is in T , then the extension of this path along Q to z satisfies condition(2) of Theorem 1 for (D,C, T ). In the other case, there is an arc wt whose initialvertex w is not in DA. We apply Lemma 1 to the triple (D,C, T ) with p = u andX the union of the vertex sets of C and C ′, concluding by (a) that w is reachable inD −A from a vertex r in X ∪ T . Assume that r is chosen so that the path from rto w in D−A has minimal length. In all cases, at least one of the three conditionsof Theorem 1 is satisfied for (D,C, T ):

if r is in C, then (1) holds;if r is in C ′, then either (1) or (3) holds;if r is in T −X , then (2) holds.

Finally, suppose that condition (3) of Theorem 3 holds for (DA, C′, TA). Then

A contains a (t, C ′)-path Q′ of type k containing u, for some vertex t in TA. Q′extends along Q to a path that ends at z and has k chords: one is zu and k − 1others are outgoing arcs at the final vertex ofQ′. If t is in T , then this path satisfiescondition (2) for (D,C, T ). In the other case, there is an arcwtwhose initial vertexw is not in DA. If w is in C, then D contains a cycle satisfying condition (1). Ifw is not in C, let B = R−(w,X) and note that B is disjoint from A; this followsfrom the fact that w is not in A. If any arc goes from C to B, then again (1) issatisfied. IfB contains any vertex in T , then (2) is satisfied. In all other cases, (b) ofLemma 2 applies with S = C, S′ = C ′, and p = w, establishing that (DB, C

′, TB)is a negative Thomassen triple of degree k. Applying the inductive hypothesis ofTheorem 2 in dual form to this triple, we consider the possibilities.

If (1) holds, thenDB contains a cycle that has k chords in the augmented digraphD∗. As before, either this cycle does not include arc zv or else DB contains a pathfrom v to z that has k chords in D. In either case, condition (1) or (3) of Theorem1 is satisfied for the triple (D,C, T ).

If (2) holds, thenB contains a (C ′, x)-path that has k chords and ends at a vertexx in TB . In this case, there is an arc xy whose terminal vertex y is not in DB .Lemma 1(b) applies to the triple (D,C, T ) with p = w and X as before, showingthat y can reach to X in D − B. Again either (1) or (3) of Theorem 1 is satisfiedfor (D,C, T ).

Finally, suppose that condition (3) of Theorem 2 holds for (DB, C′, TB). If B

contains a (C ′, w)-path that has k chords, then, by combining this path with Q′,we find again that either condition (1) or (3) of Theorem 1 is satisfied for the triple(D,C, T ). In the remaining case, B contains a (C ′, w)-path of type k. ApplyingLemma 3 to this path along withQ′, we conclude thatDB contains a cycle that has2k− 2 ≥ k chords in the augmented digraph D∗. Once again it follows that eithercondition (1) or (3) of Theorem 1 is satisfied for the triple (D,C, T ).


2.2. Induction Step for Theorem 2, Using the Present Case of Theorem1 (Same Number of Vertices in D − CD − CD − C)

Let v be any vertex in D − C. Necessarily v can reach to C in D, since the setR+(v, C) must have at least one outgoing arc. LetQ be a (v, C)-path of maximumlength and let z be the final vertex of Q. Either Q is a (v, C)-path of type k,satisfying (3) of Theorem 2, or else z dominates a vertex u in D− (C ∪Q). In thelatter case, Theorem 1 applies, with the maximality of Q implying that R+(u,Q)contains no vertex of C. We conclude that either (1) or (2) is satisfied, or else Dcontains a (v, C)-path having k chords.

2.3. Induction Step for Theorem 3, Using the Present Cases of Theo-rems 1 and 2 and the Inductive Hypotheses of Theorems 1,2, and 3

Fix a vertex v inD−C that is dominated by some vertex x ofC. We will establishfirst that either condition (1) or (2) is satisfied, or a path exists containing v as in(3), or D contains one of three special configurations shown in Figure 1. Finallywe will complete the proof by showing that the existence of any one of the threeconfigurations leads to one of the previous conclusions.

Applying Theorem 2 to the triple (D,C, T ), we find that either (1) or (2) issatisfied, or else D contains a (v, C)-path of type k. We assume that the latterholds. If v is in T , then this path satisfies condition (3) of Theorem 3; and if asecond arc goes from C to v, then Lemma 3 applies with {v} as a (C, v)-path oftype 2, providing a cycle that satisfies (1). Assuming that neither of these conditionshold, there must exist at least k−1 incoming arcs at v starting at vertices inD−C.If any such vertex q is not reachable from v inD−C, thenD contains configuration#1. Therefore, we can assume that any vertex q in D − C that dominates v isreachable from v in D − C.

Let Q be a (v, C)-path that maximizes the length of the path P defined as inconfiguration #2: The final vertex z of Q dominates a vertex r of C, and P isthe path in C starting at r and ending at x. If v is dominated by some vertex q inD− (C ∪Q), then D contains configuration #2, with the maximality of P and thefact that q is reachable from v inD−C implying thatR+(q, P ) contains no vertexof C. Therefore, we assume that all incoming arcs at v start in C or Q. Since wehave already established that only one such arc starts in C, all others (of whichthere are at least k − 1) are chords of Q. In this case, Q is a (C, z)-path of type k.

Among all (v, C)-paths that contain Q, let Q′ be one having maximal lengthand let y be its final vertex. Some arc goes from y to C, and a second outgoingarc must exist at y. If such an arc goes to C, then Lemma 3 applies with {y} asa (y, C)-path of type 2, providing a cycle that satisfies (1). Therefore, we assumethat y dominates some vertex w inD−C. If w is not inQ′, then maximality ofQ′implies that the set R+(w,Q′) contains no vertex of C. Then Theorem 1 applies,leading to the conclusion that either (1) or (2) is satisfied. In the remaining case, w


FIGURE 1. Configuration #1. There is a (v, C)-path Q of type k and a vertex q in D − Cthat dominates v and is not reachable from v in D − C.Configuration #2. There is a (v, C)-path Q and a vertex q in D− (C ∪Q) that dominates v;the final vertex z of Q dominates a vertex r of C; and R+(q, P ) contains no vertex of C,where P is the path in C starting at r and ending at x.Configuration #3. There is a (v, C)-path Q whose final vertex z dominates v, and a vertexu in D − (C ∪ Q) that is dominated by v; z dominates a vertex r of C; and R+(u, P ∪ Q)contains no vertex of C, where P is the path in C starting at r and ending at x. Also, allincoming arcs at v start in C or Q.

is in Q′ and, consequently, yw is a chord of Q′. Since Q already has k − 1 chordsthat are incoming arcs at v, it follows that either D contains a cycle satisfying (1)or else yw is one of these chords. We assume the latter, in which case yw must bezv and Q′ = Q.

Now consider outgoing arcs at v. One of these is the first arc of Q, and at leastone more exists. If a second arc goes from v to P ∪ Q, then it is a chord of thecycle C ′ formed by Q,P , and the arcs xv and zr. (Note: If r = x, then P consistsof a single vertex.) Since C ′ also has the k − 1 chords that end at v, condition(1) is satisfied. Moreover, maximality of P implies that no arc can go from v toC−P . We can assume, therefore, that v dominates some vertex u inD− (C ∪Q).In that case, D contains configuration #3, with maximality of P implying thatR+(u, P ∪Q) contains no vertex of C.

Before proceeding further with the proof, we establish one more lemma.


Lemma 4. In the situation of Lemma 2, suppose that S and S′ are the vertexsets of cycles C and C ′ that have a nonempty intersection. Then:

(1) If either DA or DB contains a cycle having k chords, each of which has atmost one endpoint in C ′, then condition (1) of Theorem 3 is satisfied for thetriple (D,C, T ).

(2a) If A contains a (t, C ′)-path having k chords for some t in TA, then eithercondition (1) or (2) of Theorem 3 is satisfied for the triple (D,C, T ).

(2b) IfB contains a (C ′, t)-path having k chords for some t in TB, then condition(1) of Theorem 3 is satisfied for the triple (D,C, T ).

Proof. (1) follows from the fact that A and B contain no vertices of C. For(2a), if t is in T , then condition (2) holds; and if t is in TA − T , then Lemma1(a) applies with X = S ∪ S′, leading to a cycle satisfying condition (1) or a pathsatisfying (2). In the case of (2b), the existence of a cycle satisfying condition (1)follows from Lemma 1(b).

Completion of the proof if D contains configuration #1: Let B denote the setR−(q, C). The condition that q is not reachable from v in D − C implies that Bcontains no vertex of Q. If B contains any vertex in T , then Q can be extendedthrough q to a path satisfying condition (3) of Theorem 3. Therefore, we assumethat B ∩ T is empty. Applying Lemma 2 with S = S′ = C and p = q, we con-clude by (b) that (DB, C, TB) is a negative Thomassen triple of degree k. Theorem2 applies in dual form to this triple, and Lemma 4 shows that we need only considerthe case in which condition (3) of Theorem 2 holds for (DB, C, TB). Then Bcontains a (C, q)-path having k chords or a (C, q)-path of type k. In the latter case,Lemma 3 applies to this path along with Q and arc qv. In either case, D containsa cycle satisfying condition (1).

Completion of the proof if D contains configuration #2: Let B denote the setR−(q, C ∪ Q) and let C ′ be the cycle formed by Q,P , and arcs xv and zr. IfB contains any vertex t in T , then Q extends back through q to a (t, C)-path Q′.Among all (t, C)-paths that contain Q′, let Q′′ be one having maximal length andlet y be its final vertex. Either Q′′ is of type k or else y dominates some vertex uin D − (C ∪ Q′′). In the first case, Q′′ satisfies condition (3) of Theorem 3. Inthe second, maximality of Q′′ implies that the set R+(u,Q′′) contains no vertex ofC. Then Theorem 1 applies to the path Q′′, leading to the conclusion that eithercondition (1) or (2) is satisfied for the triple (D,C, T ).

Now suppose that some arc goes from C − C ′ (= C − P ) to B. Let w be theterminal vertex of this arc and repeat the argument above withw replacing t. Again,eitherQ′′ is of type k or else Theorem 1 applies toQ′′. In the latter case, it is againtrue that either (1) or (2) is satisfied, since a (w,C)-path with k chords extends toa cycle satisfying (1). In the case where Q′′ is of type k,Q′′ combines with a pathP ′ in C along with two connecting arcs to form a cycle that includes the vertex x.(This requires showing that x is in P ′: We know that P ′ ends in C − P , and thecondition R+(q, P ) ∩ C = ∅ implies that P ′ starts in P .) Moreover, we choosethe connecting arc from y to P so that the length of P ′ is maximized. This cycle


satisfies condition (1) for the triple (D,C, T ): one chord is xv and k− 1 others areoutgoing arcs at y.

Having disposed of the cases above, we can assume that (DB, C′, TB) sat-

isfies the conditions in (b) of Lemma 2, applied with S = C, S′ = C ′, andp = q, establishing that (DB, C

′, TB) is a negative Thomassen triple of degree k.DB − C ′ = B contains fewer vertices than D − C, so the inductive hypothesis ofTheorem 3 in dual form applies to this triple. By Lemma 4, we need only considerthe case in which condition (3) holds. Then B contains a (C ′, w)-path of type kcontaining q and ending at a vertex w in TB . There is an arc wu whose terminalvertex u is not in DB . Moreover, u is not in C, since u is in R+(q, P ). There-fore, u is in D − X , where X is the union of the vertex sets of C and C ′. LetA = R+(u,X) and note that no arc goes from A to C − C ′, since A is containedin R+(q, P ). Therefore, (a) of Lemma 2 applies with S = C, S′ = C ′, and p = u,establishing that (DA, C

′, TA) is a positive Thomassen triple of degree k. Theinductive hypothesis of Theorem 2 applies to this triple, and by Lemma 4 we needonly consider the case in which condition (3) holds. ThenA contains a (u,C ′)-pathhaving k chords or a (u,C ′)-path of type k. In the latter case, the fact that u isnot in B implies that A and B are disjoint, so Lemma 3 can be applied to this pathtogether with arcwu and the (C ′, w)-path of type k inB. In either case,D containsa cycle satisfying condition (1).

Completion of the proof if D contain configuration #3: Let A denote the setR+(u,C ∪Q) and let C ′ be the cycle formed by the paths Q,P , and arcs xv andzr. The condition R+(u, P ∪ Q) ∩ C = ∅ implies that no arc goes from A toC − C ′. Therefore, (a) of Lemma 2 applies with S = C, S′ = C ′, and p = u,establishing that (DA, C

′, TA) is a positive Thomassen triple of degree k. Theinductive hypothesis of Theorem 2 applies to this triple, and by Lemma 4 we needonly consider the case in which condition (3) holds. If A contains a (u,C ′)-pathhaving k chords, then D contains a cycle satisfying condition (1). Therefore, weassume thatA contains a (u,C ′)-pathQ′ of type k. Let y be the final vertex of thispath. From the set of vertices of C ′ that are dominated by y, select s so that thelength of the path P ′ from s to x in C ′ is maximized. Notice that s cannot be v,since all incoming arcs at v start in C or Q. A cycle C ′′ is formed by Q′, P ′, andarcs ys, xv, and vu. Maximality of P ′ implies that C ′′ has k − 1 chords that areoutgoing arcs at y. If s is in Q, then zv is a kth chord, and C ′′ satisfies condition(1). Therefore, we assume that s is in P , in which case Q′ is a (u,C)-path of typek. This fact will be used later.

Let B denote the set R−(z, C ∪ C ′′). If B contains any vertex t in T , thenQ′ extends through arcs zv and vu to a (t, C)-path of type k, satisfying condition(3). If an arc goes from C − C ′′ to B, then Q′ extends through zv and vu toa cycle satisfying condition (1): One chord is xv and k − 1 others are outgoingarcs at y. In all other cases, (b) of Lemma 2 applies with S = C, S′ = C ′′, andp = z, establishing that (DB, C

′′, TB) is a negative Thomassen triple of degree k.Theorem 2 applies in dual form to this triple, and by Lemma 4 we can assume thatcondition (3) holds. IfB contains a (C ′′, z)-path having k chords, then such a path


extends through arc zv to a cycle satisfying condition (1). Therefore, we assumethatB contains a (C ′′, z)-pathQ′′ of type k. Letw be the first vertex ofQ′′. Ifw isdominated by some vertex of C ′′ − C, select such a vertex s′ so that the length ofthe path P ′′ in C ′′ from v to s′ is maximized. Then P ′′, Q′′, and P combine witharcs s′w, zr, and xv to form a cycle satisfying condition (1): One chord is zv andk − 1 others are incoming arcs at w.

In the remaining case, k incoming arcs at w come from P ′ or Q′′, with at leastone of these coming from P ′. ThenQ′′ is a (C, z)-path of type k. Finally, applyingLemma 3 to Q′′ along with Q′ and arcs zv and vu, we conclude that D contains acycle satisfying condition (1). This completes the proof.

2.4. Relation to Thomassen’s Proof

To explain the relation between the foregoing argument and the one given in [2], itis necessary to modify some of our definitions: Let S and T be two disjoint sets ofvertices in a digraph D such that S and D − S are nonempty. We call (D,S, T ) astrong positive Thomassen triple of degree k, if:

(1) every nonempty set of vertices in D − S has at least k outgoing arcs; and(2) every nonempty set of vertices in D − (S ∪ T ) has at least k incoming arcs.

Interchanging the terms ‘‘incoming’’ and ‘‘outgoing’’ results in the definition ofa strong negative Thomassen triple of degree k. Finally, a (v, S)-path Q is weaklyof type k, if there are at least k arcs that are either chords of Q or else start at thefinal vertex of Q and terminate in S.

The admissible triples defined in [2] are equivalent to strong positive Thomassentriples of degree 2, while co-admissible triples are equivalent to strong negativeThomassen triples of degree 2. Propositions 3.2 and 3.3 in [2] correspond roughlyto Theorems 2 and 3, respectively, in the present work. More specifically, thesetwo propositions can be restated in the following way:

Proposition 3.2. Let (D,P, T ) be a strong positive Thomassen triple of degree2, where P is a path in D, and suppose that no arcs go from P to D − P. Then atleast one of the following conditions is satisfied:

(1) D − P contains a cycle having two chords;(2) D contains a (y, P )-path having two chords, for some vertex y in T ;(3) For each vertex v in T,D contains a (v, P )-path that is weakly of type 2.

Proposition 3.3. Let (D,P, T ) be a strong positive Thomassen triple of degree2, where P is a path inD. Then at least one of the following conditions is satisfied:

(1) D contains a cycle having two chords;(2) D contains a (y, P )-path having two chords, for some vertex y in T ;(3) For each vertex u in D − P that is dominated by the final vertex of P, there

exists a (y, P )-path that is weakly of type 2 and containsu, for some vertex y in


T, or a path that satisfies one of the conditions specified in (iii) of Proposition3.3 in [2].

The somewhat complicated conditions contained in (iii) imply the existence of acycle having two chords in the augmented digraphD∗ that results by adding an arcfrom the final vertex of P to the initial vertex. It was this observation that providedthe motivation for the use of cycles in place of paths in the present work.

References

[1] D. Marcus, Spanning subgraphs of k-connected digraphs, J Comb Theory B30 (1981), 21–31.

[2] C. Thomassen, Directed cycles with two chords and strong spanning directedsubgraphs with few arcs, J Comb Theory B 66 (1996), 24–33.

Documents

Directed cycles with chords