Directed Hamilton Cycle Research Articles

Spanning cycles in random directed graphs

AbstractWe show that, in almost every ‐vertex random directed graph process, a copy of every possible ‐vertex oriented cycle will appear strictly before a directed Hamilton cycle does, except of course for the directed cycle itself. Furthermore, given an arbitrary ‐vertex oriented cycle, we determine the sharp threshold for its appearance in the binomial random directed graph. These results confirm, in a strong form, a conjecture of Ferber and Long.

Rainbow Hamiltonicity in uniformly coloured perturbed digraphs

AbstractWe investigate the existence of a rainbow Hamilton cycle in a uniformly edge-coloured randomly perturbed digraph. We show that for every $\delta \in (0,1)$ there exists $C = C(\delta ) \gt 0$ such that the following holds. Let $D_0$ be an $n$ -vertex digraph with minimum semidegree at least $\delta n$ and suppose that each edge of the union of $D_0$ with a copy of the random digraph $\mathbf{D}(n,C/n)$ on the same vertex set gets a colour in $[n]$ independently and uniformly at random. Then, with high probability, $D_0 \cup \mathbf{D}(n,C/n)$ has a rainbow directed Hamilton cycle.This improves a result of Aigner-Horev and Hefetz ((2021) SIAM J. Discrete Math.35(3) 1569–1577), who proved the same in the undirected setting when the edges are coloured uniformly in a set of $(1 + \varepsilon )n$ colours.

Counting Hamilton cycles in sparse random directed graphs

AbstractLet be the random directed graph on n vertices where each of the possible arcs is present independently with probability p. A celebrated result of Frieze shows that if then typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in is typically . We also prove a hitting‐time version of this statement, showing that in the random directed graph process, as soon as every vertex has in‐/out‐degrees at least 1, there are typically directed Hamilton cycles.

Packing Directed Hamilton Cycles Online

Consider a directed analogue of the random graph process on $n$ vertices, where the $n(n-1)$ edges are ordered uniformly at random and revealed one at a time. It is known that with high probability (w.h.p.) the first digraph in this process with both in-degree and out-degree $\geq q$ has a $q$-edge-coloring with a Hamilton cycle in each color. We show that this coloring can be constructed online, where each edge must be irrevocably colored as soon as it appears. In a similar fashion, for the undirected random graph process, we present an online $n$-edge-coloring algorithm which yields w.h.p. $q$ disjoint rainbow Hamilton cycles in the first graph containing $q$ disjoint Hamilton cycles.

On the Maximum Number of Spanning Copies of an Orientation in a Tournament

For an orientation H with n vertices, let T(H) denote the maximum possible number of labelled copies of H in an n-vertex tournament. It is easily seen that T(H) ≥ n!/2e(H), as the latter is the expected number of such copies in a random tournament. For n odd, let R(H) denote the maximum possible number of labelled copies of H in an n-vertex regular tournament. In fact, Adler, Alon and Ross proved that for H=Cn, the directed Hamilton cycle, T(Cn) ≥ (e−o(1))n!/2n, and it was observed by Alon that already R(Cn) ≥ (e−o(1))n!/2n. Similar results hold for the directed Hamilton path Pn. In other words, for the Hamilton path and cycle, the lower bound derived from the expectation argument can be improved by a constant factor. In this paper we significantly extend these results, and prove that they hold for a larger family of orientations H which includes all bounded-degree Eulerian orientations and all bounded-degree balanced orientations, as well as many others. One corollary of our method is that for any fixed k, every k-regular orientation H with n vertices satisfies T(H) ≥ (ek−o(1))n!/2e(H), and in fact, for n odd, R(H) ≥ (ek−o(1))n!/2e(H).

Robust Hamiltonicity of random directed graphs

In his seminal paper from 1952 Dirac showed that the complete graph on n≥3 vertices remains Hamiltonian even if we allow an adversary to remove ⌊n/2⌋ edges touching each vertex. In 1960 Ghouila–Houri obtained an analogue statement for digraphs by showing that every directed graph on n≥3 vertices with minimum in- and out-degree at least n/2 contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle.A natural way to generalize such results to arbitrary graphs (digraphs) is using the notion of local resilience. The local resilience of a graph (digraph) G with respect to a property P is the maximum number r such that G has the property P even if we allow an adversary to remove an r-fraction of (in- and out-going) edges touching each vertex. The theorems of Dirac and Ghouila–Houri state that the local resilience of the complete graph and digraph with respect to Hamiltonicity is 1/2. Recently, this statements have been generalized to random settings. Lee and Sudakov (2012) proved that the local resilience of a random graph with edge probability p=ω(log⁡n/n) with respect to Hamiltonicity is 1/2±o(1). For random directed graphs, Hefetz, Steger and Sudakov (2014) proved an analogue statement, but only for edge probability p=ω(log⁡n/n). In this paper we significantly improve their result to p=ω(log8⁡n/n), which is optimal up to the polylogarithmic factor.

Counting and packing Hamilton cycles in dense graphs and oriented graphs

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly cn-regular oriented graph on n vertices with c>3/8 contains (cn/e)n(1+o(1))n directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments. We also prove that every graph G on n vertices of minimum degree at least (1/2+o(1))n contains at least (1−o(1))regeven(G)/2 edge-disjoint Hamilton cycles, where regeven(G) is the maximum even degree of a spanning regular subgraph of G. This establishes an approximate version of a conjecture of Kühn, Lapinskas and Osthus.

Random directed graphs are robustly Hamiltonian

A classical theorem of Ghouila-Houri from 1960 asserts that every directed graph on n vertices with minimum out-degree and in-degree at least n/2 contains a directed Hamilton cycle. In this paper we extend this theorem to a random directed graph D(n,p), that is, a directed graph in which every ordered pair (u, v) becomes an arc with probability p independently of all other pairs. Motivated by the study of resilience of properties of random graphs, we prove that if p≫logn/n, then a.a.s. every subdigraph of D(n,p) with minimum out-degree and in-degree at least (1/2+o(1))np contains a directed Hamilton cycle. The constant 1/2 is asymptotically best possible. Our result also strengthens classical results about the existence of directed Hamilton cycles in random directed graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 345–362, 2016

Nonexistence of a pair of arc disjoint directed Hamilton cycles on line digraphs of 2-diregular digraphs

Bermond conjectured that if G is Hamilton cycle decomposable, then L(G), the line graph of G is Hamilton cycle decomposable. In this paper, we prove that, for any k > 5, there exists a directed Hamilton cycle decomposable 2-diregular digraph D of order 2k such that L(D) is not directed Hamilton cycle decomposable.

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.

Getting a Directed Hamilton Cycle Two Times Faster

Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if, at time t, instead of a random direction one is allowed to choose the orientation of et, then whether or not it is possible to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p. the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.

Large Sets of Hamilton Cycle and Path Decompositions of Complete Bipartite Graphs

In this paper, we determine the existence spectrums for large sets of Hamilton cycle and path (resp. directed Hamilton cycle and path) decompositions of ?K m, n (resp. $${\lambda K^{*}_{m,n}}$$ ).

Directed Hamilton Cycle Decompositions of the Tensor Products of Symmetric Digraphs

In this paper, the existence of directed Hamilton cycle decompositions of symmetric digraphs of tensor products of regular graphs, namely, $$(K_r \times K_s)^*,\,\,((K_r \circ \overline{K}_s) \times K_n)^*,\,\,((K_r \times K_s) \times K_m)^*,\,\,((K_r \circ \overline{K}_s) \times (K_m \circ \overline{K}_n))^*$$and $$(K_{r,r} \times (K_m \circ \overline{K}_n))^*$$, where × and ∘ denote the tensor product and the wreath product of graphs, respectively, are proved. In [16], Ng has obtained a partial solution to the following conjecture of Baranyai and Szasz [6], see also Alspach et al. [1]: If D 1 and D 2 are directed Hamilton cycle decomposable digraphs, then D 1 ∘ D 2 is directed Hamilton cycle decomposable. Ng [17] also has proved that the complete symmetric r-partite regular digraph, $$K_{r(s)}^{*} = (K_r \circ \overline{K}_s)^*$$, is decomposable into directed Hamilton cycles if and only if $$(r,s) \ne (4,1)$$or (6, 1); using the results obtained here, we give a short proof of it, when $$r \notin {4,6}$$.

K-Ordered Hamilton cycles in digraphs

Given a digraph D, let δ 0 ( D ) : = min { δ + ( D ) , δ − ( D ) } be the minimum semi-degree of D. D is k-ordered Hamiltonian if for every sequence s 1 , … , s k of distinct vertices of D there is a directed Hamilton cycle which encounters s 1 , … , s k in this order. Our main result is that every digraph D of sufficiently large order n with δ 0 ( D ) ⩾ ⌈ ( n + k ) / 2 ⌉ − 1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, Sárközy and Selkow [H. Kierstead, G. Sárközy, S. Selkow, On k-ordered Hamiltonian graphs, J. Graph Theory 32 (1999) 17–25].

Large sets of Hamilton cycle and path decompositions

The existence spectrums for large sets of Hamilton cycle decompositions and Hamilton path decompositions are completed. Also, we show that the completion of large sets of directed Hamilton cycle decompositions and directed Hamilton path decompositions depends on the existence of certain special tuscan squares. Several conjectures about special tuscan squares are posed.

Maximal sets of Hamilton cycles in [formula omitted

In this paper, we prove that there exists a maximal set of m directed Hamilton cycles in D n if and only if ⌈ n / 2 ⌉ ⩽ m ⩽ n - 1 for n ⩾ 7 .

Hamilton decompositions of directed cubes and products

Call a directed graph G ↔ symmetric if it is obtained from an undirected graph G by replacing each edge of G by two directed edges, one in each direction. We will show that if G has a Hamilton decomposition with certain additional structure, then G ↔ × C ↔ n × K ↔ 2 has a directed Hamilton decomposition. In particular, it will follow that the bidirected cubes Q ↔ 2 m + 1 for m ⩾ 2 are decomposable into 2 m + 1 directed Hamilton cycles and that a product of cycles C ↔ n 1 × ⋯ × C ↔ n m × K ↔ 2 is decomposable into 2 m + 1 directed Hamilton cycles if n i ⩾ 3 and m ⩾ 2 .

Hamilton cycles in the union of random permutations

We prove that with high probability, two random permutations contain an undirected Hamilton cycle and three random permutations almost always contain a directed Hamilton cycle. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 83–94, 2001

The union of two random permutations does not have a directed Hamilton cycle

The union of two random permutations does not have a directed Hamilton cycle

Expected numbers at hitting times

We determine exactly the expected number of hamilton cycles in the random graph obtained by starting with n isolated vertices and adding edges at random until each vertex degree is at least two. This complements recent work of Cooper and Frieze. There are similar results concerning expected numbers, for example, of perfect matchings, spanning trees, hamilton paths, and directed hamilton cycles.