Cycles In Digraphs Research Articles

Two disjoint cycles in digraphs

AbstractBermond and Thomassen conjectured that every digraph with minimum outdegree at least contains vertex disjoint cycles. So far the conjecture was verified for . Here we generalise the question asking for all outdegree sequences which force vertex disjoint cycles and give the full answer for .

Partition and Disjoint Cycles in Digraphs

Partition and Disjoint Cycles in Digraphs

4-Free Strong Digraphs with the Maximum Size

Directed cycles in digraphs are useful in embedding linear arrays and rings, and are suitable for designing simple algorithm with low communication costs in parallel computer systems, thus the existence of directed cycles on digraphs has been largely investigated. Let [Formula: see text], [Formula: see text] be integers. Bermond et al. [Journal of Graph Theory 4(3) (1980) 337–341] proved that if the size of a strong digraph [Formula: see text] with order [Formula: see text] is at least [Formula: see text], then the girth of [Formula: see text] is no more than [Formula: see text]. Consequently, when [Formula: see text] is a 4-free strong digraph with order [Formula: see text], which means that every directed cycle in [Formula: see text] has length at least [Formula: see text], then the maximum size of [Formula: see text] is [Formula: see text]. In this paper, we mainly give the structural characterizations for all 4-free strong digraphs of order [Formula: see text] whose arc number exactly is [Formula: see text].

Oriented Cycles in Digraphs of Large Outdegree

In 1985, Mader conjectured that for every acyclic digraph F there exists K = K(F) such that every digraph D with minimum out-degree at least K contains a subdivision of F. This conjecture remains widely open, even for digraphs F on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomassé studied special cases of Mader’s problem and made the following conjecture: for every ℓ ≥ 2 there exists K = K(ℓ) such that every digraph D with minimum out-degree at least K contains a subdivision of every orientation of a cycle of length ℓ.We prove this conjecture and answer further open questions raised by Aboulker et al.

Subdivisions of four blocks cycles in digraphs with large chromatic number

A cycle with four blocks C(k1,k2,k3,k4) is an oriented cycle formed of four blocks of lengths k1,k2,k3 and k4 respectively. We conjecture that for every positive integers k1,k2,k3,k4, there is an integer g(k1,k2,k3,k4) such that every strongly connected digraph with chromatic number greater than g(k1,k2,k3,k4) contains a subdivision of C(k1,k2,k3,k4). As evidence, we prove this conjecture for k2=k3=k4=1.

Algebraic Algorithms for Even Circuits in Graphs

We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.

On the Number of Vertex-Disjoint Cycles in Digraphs

Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. This conjecture is famous as ...

Subdivisions of oriented cycles in digraphs with large chromatic number

AbstractAn oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle C, there are digraphs containing no subdivision of C (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any C a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of C. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out‐degree 2 and two vertices have in‐degree 2).

Cycles with two blocks in k‐chromatic digraphs

AbstractLet k and ℓ be positive integers. A cycle with two blocks is a digraph obtained by an orientation of an undirected cycle, which consists of two internally (vertex) disjoint paths of lengths at least k and ℓ, respectively, from a vertex to another one. A problem of Addario‐Berry, Havet and Thomassé [J. Combin. Theory Ser. B 97 (2007), 620–626] asked if, given positive integers k and ℓ such that , any strongly connected digraph D containing no has chromatic number at most . In this article, we show that such digraph D has chromatic number at most , improving the previous upper bound of Cohen et al. [Subdivisions of oriented cycles in digraphs with large chromatic number, to appear]. We also show that if in addition D is Hamiltonian, then its underlying simple graph is ‐degenerate and thus the chromatic number of D is at most , which is tight.

Pfaffians and nonintersecting paths in graphs with cycles: Grassmann algebra methods

After recalling the definition of Grassmann algebra and elements of Grassmann–Berezin calculus, we use the expression of Pfaffians as Grassmann integrals to generalize a series of formulas relating generating functions of paths in digraphs to Pfaffians. We start with the celebrated Lindström–Gessel–Viennot formula, which we derive in the general case of a graph with cycles. We then make further use of Grassmann algebraic tools to prove a generalization of the results of Stembridge [13]. Our results, which are applicable to graphs with cycles, are formulated in terms of systems of nonintersecting paths and nonintersecting cycles in digraphs.

Vertex-disjoint cycles in bipartite tournaments

Let k≥2 be an integer. Bermond and Thomassen [Bermond J. C., Thomassen, C., Cycles in digraphs a survey, Journal of Graph Theory 5(1) (1981) 1–43] conjectured that every digraph D with δ+(D)≥2k−1 contains at least k vertex-disjoint cycles. In this work we prove that every bipartite tournament with minimum out-degree at least 2k−2 and minimum in-degree at least one contains k vertex-disjoint cycles of length four, whenever k≥3. Finally, we show that every bipartite tournament with minimum degree at least (3k−1)/2 contains k vertex-disjoint cycles of length four.

On longest non-Hamiltonian cycles in digraphs with the conditions of Bang-Jensen, Gutin and Li

Let D be a strongly connected directed graph of order n≥4. In Bang-Jensen et al. (1996), (J. of Graph Theory 22 (2) (1996) 181–187), J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If (∗)d(x)+d(y)≥2n−1 and min{d(x),d(y)}≥n−1 for every pair of non-adjacent vertices x,y with a common in-neighbour or (∗∗)min{d+(x)+d−(y),d−(x)+d+(y)}≥n for every pair of non-adjacent vertices x,y with a common in-neighbour or a common out-neighbour, then D is Hamiltonian. In this paper we show that: (i) if D satisfies condition (∗) and the minimum semi-degree of D at least two or (ii) if D is not directed cycle and satisfies condition (∗∗), then either D contains a cycle of length n−1 or n is even and D is isomorphic to the complete bipartite digraph or to the complete bipartite digraph minus one arc.

Arc‐Disjoint Directed and Undirected Cycles in Digraphs

AbstractThe dicycle transversal number of a digraph D is the minimum size of a dicycle transversal of D, that is a set of vertices of D, whose removal from D makes it acyclic.An arc a of a digraph D with at least one cycle is a transversal arc if a is in every directed cycle of D (making acyclic). In [3] and [4], we completely characterized the complexity of following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in its underlying undirected graph such that . It turns out that the problem is polynomially solvable for digraphs with a constantly bounded number of transversal vertices (including cases where ). In the remaining case (allowing arbitrarily many transversal vertices) the problem is NP‐complete. In this article, we classify the complexity of the arc‐analog of this problem, where we ask for a dicycle B and a cycle C that are arc‐disjoint, but not necessarily vertex‐disjoint. We prove that the problem is polynomially solvable for strong digraphs and for digraphs with a constantly bounded number of transversal arcs (but possibly an unbounded number of transversal vertices). In the remaining case (allowing arbitrarily many transversal arcs) the problem is NP‐complete.

Arbitrary Orientations of Hamilton Cycles in Digraphs

Let $n$ be sufficiently large and suppose that $G$ is a digraph on $n$ vertices where every vertex has in- and outdegree at least $n/2$. We show that $G$ contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is $n/2+1$. Our result is best possible and improves on an approximate result by Häggkvist and Thomason.

On 2‐Colorings of Hypergraphs

AbstractIn this article, we continue the study of 2‐colorings in hypergraphs. A hypergraph is 2‐colorable if there is a 2‐coloring of the vertices with no monochromatic hyperedge. Let denote the class of all k‐uniform k‐regular hypergraphs. It is known (see Alon and Bregman [Graphs Combin. 4 (1988) 303–306] and Thomassen [J. Amer. Math. Soc. 5 (1992), 217–229] that every hypergraph is 2‐colorable, provided . As remarked by Alon and Bregman the result is not true when , as may be seen by considering the Fano plane. Indeed there are several constructions for building infinite families of hypergraphs in that are not 2‐colorable. Our main result in this paper is a strengthening of the above results. For this purpose, we define a set X of vertices in a hypergraph H to be a free set in H if we can 2‐color such that every edge in H receives at least one vertex of each color. We conjecture that for , every hypergraph has a free set of size in H. We show that the bound cannot be improved for any and we prove our conjecture when . Our proofs use results from areas such as transversal in hypergraphs, cycles in digraphs, and probabilistic arguments.

Notes on a conjecture of Manoussakis concerning Hamilton cycles in digraphs

In 1992, Manoussakis conjectured that a strongly 2-connected digraph D on n vertices is hamiltonian if for every two distinct pairs of independent vertices x, y and w, z we have d(x)+d(y)+d(w)+d(z)≥4n−3. In this note we show that D has a Hamilton path, which gives an affirmative evidence supporting this conjecture.

Directed Hamilton Cycles in Digraphs and Matching Alternating Hamilton Cycles in Bipartite Graphs

In 1972, Woodall raised the following Ore-type condition for directed Hamilton cycles in digraphs: Let $D$ be a digraph. If for every vertex pair $u$ and $v$, where there is no arc from $u$ to $v$, we have $d^+(u)+d^-(v)\geq |D|$, then $D$ has a directed Hamilton cycle. By a correspondence between bipartite graphs and digraphs, the above result is equivalent to the following result of Las Vergnas: Let $G = (B,W)$ be a balanced bipartite graph. If for any $b \in B$ and $w \in W$, where $b$ and $w$ are nonadjacent, we have $d(w) +d(b) \geq |G|/2 + 1$, then every perfect matching of $G$ is contained in a Hamilton cycle. The lower bounds in both results are tight. In this paper, we reduce both bounds by $1$ and prove that the conclusions still hold, with only a few exceptional cases that can be clearly characterized.

Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree greater than $${\frac{1}{2}n + 7\sqrt{n}/3}$$ contains an anti-directed hamiltonian cycle. In addition, we show that D contains anti-directed cycles of all possible (even) lengths when n is sufficiently large and has minimum in- and out-degree at least $${(1/2+ \epsilon)n}$$ for any $${\epsilon > 0}$$ .

Destroying cycles in digraphs

For a simple directed graph G with no directed triangles, let β(G) be the size of the smallest subset X ∈ E(G) such that G\X has no directed cycles, and let γ(G) denote the number of unordered pairs of nonadjacent vertices in G. Chudnovsky, Seymour, and Sullivan showed that β(G) ≤ γ(G), and conjectured that β(G) ≤ $\tfrac{{\gamma (G)}} {2}$. In this paper we prove that β(G)<0.88γ(G).

A Semiexact Degree Condition for Hamilton Cycles in Digraphs

We show that for each $\beta > 0$, every digraph G of sufficiently large order n whose outdegree and indegree sequences $d_1^+ \leq \cdots \leq d_n^+$ and $d_1^- \leq \cdots \leq d_n^-$ satisfy $d_i^+, d_i^- \geq \min{\{i + \beta n, n/2\}}$ is Hamiltonian. In fact, we can weaken these assumptions to (i) $d_i^+ \geq \min{\{i + \beta n, n/2\}}$ or $d^-_{n - i - \beta n} \geq n-i$, (ii) $d_i^- \geq \min{\{i + \beta n, n/2\}}$ or $d^+_{n - i - \beta n} \geq n-i$, and still deduce that G is Hamiltonian. This provides an approximate version of a conjecture of Nash-Williams from 1975 and improves a previous result of Kühn, Osthus, and Treglown.