One of the most well-known conjectures concerning Hamiltonicity in graphs asserts that any sufficiently large connected vertex transitive graph contains a Hamilton cycle. In this form, it was first written down by Thomassen in 1978, inspired by a closely related conjecture due to Lovász from 1969. It has been attributed to several other authors in a survey on the topic by Witte and Gallian in 1984. The analogous question for vertex transitive digraphs has an even longer history, having been first considered by Rankin in 1946. It is arguably more natural from the group-theoretic perspective underlying this problem in both settings. Trotter and Erdős proved in 1978 that there are infinitely many connected vertex transitive digraphs which are not Hamiltonian. This left open the very natural question of how long a directed cycle one can guarantee in a connected vertex transitive digraph on $n$ vertices. In 1981, Alspach asked if the maximum perimeter gap (the gap between the circumference and the order of the digraph) is a growing function in $n$. We answer this question in the affirmative, showing that it grows at least as fast as $(1-o(1)) \ln n$. On the other hand, we prove that one can always find a directed cycle of length at least $Ω(n^{1/3})$, establishing the first lower bound growing with $n$, providing a directed analogue of a famous result of Babai from 1979 in the undirected setting.
Finding long paths and cycles in graphs is one of the most classical directions of study in graph theory. Perhaps the most famous instance of this general direction is the question of finding the longest possible cycle, namely one that traverses all the vertices. Such a cycle is called a Hamilton cycle, and a graph containing it is said to be Hamiltonian. Hamiltonicity is a very classical and extensively studied graph property. In general, it is a hard problem to decide if a given graph is Hamiltonian. In fact, this is one of Karp's famous 21 NP-hard problems [9], and is often used to establish hardness of other computational problems. This goes a long way towards explaining why there are so many interesting results establishing sufficient conditions for Hamiltonicity. The simplest one, featured in essentially every introductory course on graph theory, is Dirac's theorem from 1952, which states that any graph with minimum degree at least n 2 is Hamiltonian. A major downside of Dirac's theorem is that it only applies for very dense graphs, leading to a more challenging question of finding natural graph properties that would force Hamiltonicity even for much sparser graphs. Perhaps the most intriguing candidate, first considered in the 1960s, is symmetry. This idea first appeared in a communication by Lovász [14] from 1969, where he conjectured that any connected vertex transitive graph must contain a Hamilton path. Thomassen (see [2]) refined this conjecture in 1978 by asserting that any sufficiently large connected vertex transitive graph is Hamiltonian.
These conjectures have attracted an immense amount of work over the years, with multiple surveys on the topic [1,5,22,28] being written, starting as early as the 1980s. Despite this attention, both conjectures remain widely open and most of what is known concerns various additional assumptions under which the conjectures hold. On the other hand, Babai [2] already in 1979 initiated a very general direction of attack, namely of trying to at least find a long cycle in every connected vertex transitive graph (without any additional assumptions). In particular, he proved that an n-vertex connected vertex transitive graph always has a cycle of length Ω( √ n). In recent years, there have been several improvements over this result, all making interesting connections to a number of other interesting graph theoretic problems [6,7,15,21] leading to the current state of the art of Ω(n 9/14 ) proved in [21].
In this paper, we are interested in the directed analogue of this problem. Namely, how long a cycle can we find in any n-vertex connected vertex transitive digraph? This question is even more classical, owing to the fact that in the arguably most interesting instance of the problem, namely that of Cayley digraphs, the directed variant is more natural 1 and translates to a natural group rearrangement problem. Indeed, the two oldest papers to consider this problem started from this group theoretic question and translated it to the Cayley digraph instance of our problem. These were a 1946 paper by Rankin [23], and an independent 1959 paper by Rapaport-Strasser [24]. Both of these works attribute their motivation to Campanology (as well as the “knight tour” problem in the latter case), where various instances of this problem have been solved in practice, more than a century before (see [23,Section 4] or [27,Chapter 15] for more details).
The directed analog of Thomassen’s conjecture was first disproved2 by Trotter and Erdős [26] in 1978, who exhibited an infinite family of connected vertex transitive digraphs without Hamilton cycles. Motivated by this result, Alspach asked already in 1981 whether such graphs need to be at least “nearly” Hamiltonian. Here, to formalize this question, we can use the perimeter gap, defined as the difference between the number of vertices and the length of the longest directed cycle in a digraph, as a measure of how far from Hamiltonian a graph is. In particular, Alspach (Question 7 in [1]) asked if for any constant C there exists a connected vertex transitive digraph with perimeter gap larger than C. We answer this question in the affirmative in the following quantitative form. Theorem 1.1. For infinitely many natural numbers n, there exists a connected vertex transitive digraph on n vertices with perimeter gap at least (1 -o(1)) ln(n).
Given this result, the question of how long of a cycle we can actually guarantee, raised by Babai in the undirected case already in 1979, becomes even more natural in the directed case. Here, it was not even known whether one can guarantee a cycle of length growing with n. We prove such a result, establishing a directed analog of Babai’s result from 1979. Theorem 1.2. In any connected vertex transitive digraph D on n ≥ 2 vertices there is a directed cycle of length at least Ω(n 1/3 ).
We note that neither Babai’s result nor the aforementioned recent results improving on it extend to the directed case.
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