In the closed version of the snake-in-the-box problem, a snake is a chordless cycle in a 
-hypercube graph. Every graph
edge joining two of its vertices must belong
to the graph cycle. Its length is its number of edges. Klee (1970) discussed the maximum length 
of such a closed snake and earlier bounds of Danzer and
Klee (1967) and Douglas (1969).
Modern terminology reserves snake for an induced graph path, whose nonconsecutive vertices are not adjacent in the hypercube graph (Allison and Paulusma 2016). The two versions have different maxima.
For closed snakes, a lower bound is
 |
(1)
|
for 
(Abbott and Katchalski 1991, Allison
and Paulusma 2016). Snevily (1994) proved the upper bound
 |
(2)
|
for 
(Lukito and van Zanten 2001).
Snevily (1994) also conjectured the upper bound
 |
(3)
|
for 
.
The exact closed-snake maxima 
for 
, 2, ... are 0, 4, 6, 8, 14, 26, 48,
96, ... (OEIS A000937). In particular, the
one-dimensional hypercube graph has no graph
cycle. The corresponding open-snake maxima, measured in edges,
for 
, 2, ... are 1, 2, 4, 7, 13, 26, 50,
98, ... (OEIS A099155, Allison and Paulusma
2016).
in the 
-Cube." J. Combin. Th. 6,
323-339, 1969.Emelianov, P. "Snake-in-the-Box." 
-Dimensional Cube." Mat. Zametki 6, 309-319,
1969.Guy, R. K. "Monthly Unsolved Problems, 1969-1987."
Amer. Math. Monthly 94, 961-970, 1987.Guy, R. K.
"Unsolved Problems Come of Age." Amer. Math. Monthly 96,
903-909, 1989.Guy, R. K. and Nowakowski, R. J. "Monthly
Unsolved Problems, 1969-1995." Amer. Math. Monthly 102, 921-926,
1995.Kautz, W. H. "Unit-Distance Error-Checking Codes."
IRE Trans. Elect. Comput. 7, 177-180, 1958.Klee, V. "What
Is the Maximum Length of a 
-Dimensional
Snake?" Amer. Math. Monthly 77, 63-65, 1970.Lukito,
A. and van Zanten, A. J. "A New Non-Asymptotic Upper Bound for Snake-in-the-Box
Codes." J. Combin. Math. Combin. Comput. 39, 147-156, 2001. 
-Dimensional Cube." Diskret. Analiz. 45,
1987.