Does Dirac's theorem on Hamiltonian cycles only apply for undirected graphs? If the theorem applied to directed graphs, the graph with the following adjacency list should have a Hamiltonian cycle, but I can't find one.

$0$: $1$ $2$ $3$

$1$: $2$ $3$

$2$: $1$ $3$

$3$: $1$ $2$

  • $\begingroup$ Yes Dirac Theorem is for undirected graphs, directed variation is this(is not by dirac): If $G$ is a strongly connected graph and $\delta^+(G)+\delta^-(G) \ge n$. $\delta^+(G)$ is minimum out degree and $\delta^-(G)$ is minimum in degree of $G$. $\endgroup$
    – Saeed
    Commented Apr 23, 2014 at 20:56

1 Answer 1


While Dirac's theorem wasn't intended/proved for directed graphs, it was extended to directed graphs by GHOUILA-HOURI in 1960 ("Une condition suffisante d’existence d'un circuit hamiltonien").

Their requirements, however, were that each vertex should have both out-degree and in-degree of at least $n/2$, which is not the case in your graph.

One of the original conditions of GHOUILA-HOURI, as Saeed mentioned, is that the graph is strongly connected. In fact, the degree conditions are strong enough for only diameter $\leq 2$ graphs to be considered.


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