Long cycles in graphs and digraphs

Lech Adamus 1,2

Faculty of Applied MathematicsAGH University of Science and Technology,

Cracow, Poland

Abstract

Consider the following problem (solved by Woodall): given p > |G|2 , find the mini-

mum size of a graph G guaranteeing the existence of a cycle of length p.We prove that a balanced bipartite graph of order 2n and size greater than

n(n − k − 1) + k + 1 contains a cycle of length 2n − 2k, where n ≥ 12k2 + 3

2k + 4.In the paper we also formulate and give a solution to an analogous problem for

digraphs.

Keywords: bipartite graph, digraph, long cycle, orientation of cycle, almostsymmetric cycle

1 Long cycles in simple graphs

We consider only finite graphs without loops and multiple edges. Our termi-nology and notation are standard (see, e. g. [5]).

Let G be a graph with vertex set V (G) and edge set E(G). Denote by |G|the number of vertices in G, and by ||G|| the number of edges in G.

1 The research was partially supported by the AGH University of Science and Technologygrant No 11 420 04.2 Email: ladamus@wms.mat.agh.edu.pl

Electronic Notes in Discrete Mathematics 24 (2006) 3–7

1571-0653/$ – see front matter © 2006 Published by Elsevier B.V.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2006.06.002

In 1960 Ore proved the following

Theorem 1.1 ([7]) Let G be a graph with |G| = n and||G|| ≥ 1

2(n − 1)(n − 2) + 2. Then G contains a hamiltonian cycle.

It is natural to consider the following problem: for any pair of integers nand k, find the minimum integer f(n, k), such that every graph of order n andsize at least f(n, k) contains a cycle of length n − k, where k ≤ n − 3.

Woodall found the number f(n, k) and, what’s more, proved that a graphof size at least f(n, k) contains cycles of all lengths up to n − k.

Theorem 1.2 ([11]) Let G be a graph with |G| = n ≥ 2k + 3 and

||G|| ≥⎛⎝ n − k − 1

2

⎞⎠ +

⎛⎝ k + 2

2

⎞⎠ + 1.

Then G contains a cycle of length p for each p such that 3 ≤ p ≤ n − k.

We solve an analogous problem for balanced bipartite graphs. Moreover,we characterize extremal graphs for this problem.

Let G = (X, Y ; E(G)), where |X| = |Y | = n, be a balanced bipartite graph oforder 2n and color classes X and Y . We prove the following

Theorem 1.3 ([1]) Let G = (X, Y ; E(G)), where |X| = |Y | == n ≥ 1

2k2 + 3

2k + 4 and ||G|| ≥ f(n, k) = n(n− k − 1) + k + 1. Then G con-

tains cycles of all even lengths not greater than 2n−2k, unless ||G|| = f(n, k)and G is isomorphic to a graph of family G= {G = (X, Y ; E(G)), whereX = A ∪ B : A = {x ∈ X : d(x) = n}, B = {x ∈ X : d(x) = 1}, |B| = k + 1}(then G does not contain a cycle of length 2n − 2k).

We expect that the condition on the order in Theorem 1.3 may be re-laxed, namely it suffices to assume that n ≥ 2k + 2. The conjecture is truefor k = 0 [8] and for k = 1 (as a corollary to a theorem by Bagga andVarma [3]).

2 Almost symmetric cycles in digraphs

The similar problem is studied for digraphs. Let us start with some prepara-tion.

L. Adamus / Electronic Notes in Discrete Mathematics 24 (2006) 3–74

2.1 Terminology

With some exceptions specified below, we follow the standard terminologyof [4]. For a digraph D, we denote by V (D) the vertex set of D, by A(D) thearc set of D, by |D| the number of vertices in D, and by ||D|| the number ofarcs in D.

For a digraph D the associated graph G(D) is the undirected graph withthe same vertex set as D such that the edges of G(D) correspond to the2-cycles (symmetric arcs) of D.

Let D1 and D2 be vertex disjoint digraphs. By D1 � D2 we denotethe family of digraphs D such that V (D) = V (D1) ∪ V (D2) andfor every x, y ∈ V (D) : (x → y) ∈ A(D) if and only if

(1) x, y ∈ V (Di) and (x → y) ∈ A(Di) (i = 1, 2) or

(2) x ∈ V (Di), y ∈ V (Dj), i �= j and (y → x) /∈ A(D) (the vertices of V (D1)are joined with the vertices of V (D2) by antisymmetric arcs).

We define the digraph D1 ∗ D2 with V (D1 ∗ D2) = V (D1) ∪ V (D2) and thearc set consisting of the arcs of D1 and D2, together with all the arcs betweenD1 and D2. Similarly, we define the digraph D1 → D2 with V (D1 → D2) =V (D1) ∪ V (D2) and the arc set consisting of the arcs of D1, D2, and all arcsfrom D1 to D2. Let us denote by Hn,k the family K∗

n−k−1 � K∗k+1 where K∗

m

denotes a complete symmetric digraph with m vertices.

A sequence ε = (ε1, ..., εk), where εi ∈ {−1, 1}, 1 ≤ i ≤ k, is calledthe orientation of a cycle C = (x1, ..., xk, x1) of D if εi = 1 implies(xi → xi+1) ∈ A(Di) and εi = −1 implies (xi+1 → xi) ∈ A(Di) for everyi(modk). Then C is a realization of ε in D. Any realization in a digraph D ofthe orientation ε = (ε1, ..., εk) is called a strong cycle if εiεi+1 = 1. C∗

k denotesa symmetric cycle, that is a cycle with symmetric arcs only, and C∗′

k denotesC∗

k minus one arc, which we call an almost symmetric cycle of length k .

2.2 Results

Lewin [6] showed that every strong digraph with n vertices and at least(n− 1)(n− 2) + 3 arcs contains a strong hamiltonian cycle. In fact, as provedby Wojda [9], with some exceptions, every digraph with n ≥ 9 vertices and atleast (n−1)(n−2)+2 arcs contains all orientations of a hamiltonian cycle. Toobtain this result Wojda used symmetric and almost symmetric hamiltoniancycles.

It is natural to investigate an analogous problem for long but not hamil-tonian cycles. For symmetric and almost symmetric cycles we have

L. Adamus / Electronic Notes in Discrete Mathematics 24 (2006) 3–7 5

Theorem 2.1 ([2]) Let D be a digraph with |D| = n, n ≥ 7, n ≥ 52k +6, and

||D|| ≥ f(n, k), where f(n, k) = (n − k − 1)(n − 1) + k(k + 1). Then:

(1) D contains all symmetric cycles C∗p for 3 ≤ p ≤ n − k − 2,

(2) D contains an almost symmetric cycle C∗′n−k−1; moreover, if

|D| ≥ 3k + 6 then D contains a symmetric cycle C∗n−k−1,

(3) D contains an almost symmetric cycle C∗′n−k unless

(3a) D is one of the digraphs from Hn,k,(3b) n = 3k + 4 and D = T2k+3 ∗ K∗

k+1,(3c) n = 3k + 2 and D = T2k+2 ∗ K∗

k ,where Tp is a tournament of order p.

In the proof of Theorem 2.1, the authors apply Theorem 1.2 to the asso-ciated graph G(D) of a digraph D.

Theorem 2.1 implies the result for all orientations of long cycles:

Corollary 2.2 ([2]) Let D be a digraph. Suppose that |D| = n, n ≥ 7,n ≥ 5

2k + 6 and ||D|| ≥ f(n, k). Then D contains every orientation of

a cycle of length n − k except the strong one in case D = K∗n−k−1 → K∗

k+1

or D = K∗k+1 → K∗

n−k−1.

Wojda and Wozniak investigated the existence of almost symmetric hamil-tonian cycles in balanced bipartite digraphs.

Let D = (X,Y ; A(D)), where |X| = |Y | = n, be a balanced bipartite digraphof order 2n and color classes X and Y .

Theorem 2.3 ([10]) Let D = (X,Y ; A(D)), where |X| = |Y | = n, and||D|| ≥ 2n2 − 2n + 3. Then D contains an almost symmetric hamiltoniancycle unless there is in D a vertex which is not incident to any symmetric arcof D.

It is interesting to study the problem of the existence of symmetric andalmost symmetric cycles of even lengths in a balanced bipartite digraph D.Some results were obtained, similarly as in the proof of Theorem 2.1, byapplying Theorem 1.3 to the associated graph G(D).

References

[1] Adamus, L., Long cycles in balanced bipartite graphs, manuscript.

L. Adamus / Electronic Notes in Discrete Mathematics 24 (2006) 3–76

[2] Adamus, L., and A. P. Wojda, Almost symmetric cycles in large digraphs,Graphs and Combinatorics, to appear.

[3] Bagga, K. S., and B. N. Varma, Bipartite graphs and degree conditions, GraphTheory, Combinatorics, Algorithms, and Applications, (ed. Y.Alavi, F.Chung,R.Graham, D.Hsu), Siam 1991, 564-573.

[4] Bermond, J. C., and C. Thomassen, Cycles in digraphs – a survey, J. GraphTheory 5 (1981), 1-43.

[5] Diestel, R., ”Graph Theory,” Springer-Verlag, New York, 1997.

[6] Lewin, M., On maximal circuits in directed graphs, J. Combinatorial Theory B18 (1975), 175-179.

[7] Ore, O., Note on Hamilton circuits, Am. Math. Monthly. 67 (1960), 55.

[8] Schmeichel, E., and J. Mitchem, Pancyclic and bipancyclic graphs. A Survey,Graphs and Applications. Proceedings of the First Colorado Symposium onGraph Theory (eds. F. Harary and J.S. Maybee), Wiley, New York (1985),271-278.

[9] Wojda, A. P., Orientations of hamiltonian cycles in large digraphs, J. GraphTheory 10 (1986), 211-218.

[10] Wojda, A. P., and M. Wozniak, Orientations of hamiltonian cycles in bipartitedigraphs, Colloquia Mathematica Societatis Janos Bolyai 60. Sets, Graphs andNumbers (1991), 719-726.

[11] Woodall, D. R., Sufficient conditions for circuits in graphs, Proc. London Math.Soc. 24 (1972), 739-755.

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