Long cycles in vertex transitive digraphs

Abstract

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.

Figures (2)

• Figure 1.1: A family of strongly $2$-connected $2$-regular digraphs such that any longest directed cycle has length four, while the maximum length of a directed path is not bounded. In this family, the number of vertex-disjoint longest directed cycles can be arbitrarily large.
• Figure 2.1: Illustration of the setup in the proof of \ref{['lem:long-cycle-in-cycle-graphs']}. Dashed lines depict pairs of vertices at distance up to one. $L'\cup R'=S'$ is a geodesic path, and so is $Q$. This implies that $d(w,a)$ is very close to $d(w,a')$ and $d(w,b)$ to $d(w,b')$. The argument now relies on the fact that $d(a',b')$ is roughly the sum of these two distances since $a',b'$ are both vertices of the geodesic path $S'$, but also by going through the $a$-$b$ segment, we can find a path of distance roughly the difference between these two distances. The conclusion is that one of these distances needs to be very small.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• proof : Proof of Lemma \ref{['lem:transitive-implies-expander']}
• Lemma 2.4
• proof
• Theorem 2.5