J. London Math. Soc. (2) 50, (1994), 465–476.
LONG SNAKES IN POWERS OF THE COMPLETE GRAPH
WITH AN ODD NUMBER OF VERTICES
Jerzy Wojciechowski
Abstract. In [5] Abbott and Katchalski ask if there exists a constant c > 0 such that for every
d ≥ 2 there is a snake (cycle without chords) of length at least c3d in the product of d copies of the
complete graph K3 . We show that the answer to the above question is positive, and that in general
for any odd integer n there is a constant cn such that for every d ≥ 2 there is a snake of length at
least cn nd in the product of d copies of the complete graph Kn .
1991 Mathematics Subject Classification 05C35, 05C38
1
1. Introduction
By a path in a graph G we mean a sequence of distinct vertices of G with every pair of
consecutive vertices being adjacent. A path will be called closed if its first vertex is adjacent
to the last one. By a chord of a path P in a graph G we mean an edge of G joining two
nonconsecutive vertices of P . If e is a chord in a closed path P , then e is called proper if it
is not the edge joining the first vertex of P to its last vertex. Note that a proper chord of a
closed path corresponds to the standard notion of a chord in a cycle. A snake in a graph G
is a closed path in G without proper chords, and an open snake in G is a path in G without
chords.
If G and H are graphs, then the product of G and H is the graph G×H with V (G)×V (H)
as the vertex set and (g1 , h1 ) adjacent to (g2 , h2 ) if either g1 g2 ∈ E(G) and h1 = h2 , or else
g1 = g2 and h1 h2 ∈ E(H). Let Knd be the product of d copies of the complete graph Kn ,
n ≥ 2, d ≥ 1. It will be convenient to think of the vertices of Knd as d-tuples of n-ary digits,
i.e. the elements of the set {0, 1, . . . , n − 1}, with edges between two d-tuples differing at
exactly one coordinate.
Let S(Knd ) be the length of the longest snake in Knd . The problem of estimating the
value of S(K2d ) (known also as the snake-in-the-box problem) has an extensive literature.
See [2], [6], [8], and the references in these papers. Evdokimov [6] was the first to prove that
for some constant c > 0,
S(K2d ) ≥ c2d .
(1)
The constant c he got in his proof is very small (c = 2−11 ). Other shorter proofs of (1)
and with larger values of the constant c have been given by Abbott and Katchalski [2] and
Wojciechowski [8]. The best lower bound for S(K2d ) has been given by Abbott and Katchalski
[4]. They show that (1) holds with c =
77
256
= 0.300781 . . .. In [9] Wojciechowski proves a
result of a different but related nature, namely, that for every d ≥ 2, K2d can be decomposed
into 16 vertex disjoint snakes.
2
When n ≥ 3, the existence of snakes of length cnd , where c is a positive constant
independent of d, is known only for n even. Abbott and Katchalski [5] proved that if n ≡
0 mod 4, then for d ≥ 3
S(Knd ) ≥
n d−1
2
S(K2d−1 ).
(2)
It follows from (1) and (2) that in the case when n ≡ 0 mod 4 the following theorem is true.
Theorem 1. For any even integer n ≥ 2, there exists a constant cn > 0 such that
S(Knd ) ≥ cn nd ,
(3)
for every d ≥ 2.
When n ≡ 2 mod 4, the validity of Theorem 1 follows from the remark in [5] that the
construction used in the proof of (2) can be modified to give long snakes in the case when
n ≡ 2 mod 4.
For n odd, the best known lower bounds on S(Knd ) are not as good as for even n. Abbot
and Dierker [1] proved that
S(Knd )
≥4
j n kd−1
2
+2
d−2 j kk
X
n
k=1
2
for n ≥ 2, d ≥ 2,
and
S(Knd )
≥
2nd−`−1
2nd−`
if 2` < d < 2`+1
if d = 2` .
It follows that for every n ≥ 2 there is a constant λn > 0 such that
S(Knd ) ≥
1 d
n .
dλn
Abbott and Katchalski [5] ask whether there exists a constant c3 > 0 such that (3) holds for
n = 3. In this paper we give a positive answer to this question, and what is more, we show
that the following general result is true.
3
Theorem 2. For any odd integer n ≥ 3, there exists a constant cn > 0 such that
S(Knd ) ≥ cn nd ,
for every d ≥ 2.
Actually, we prove a stronger result.
Theorem 3. For any odd integer n ≥ 3, and any d ≥ 5
S(Knd ) ≥ 2(n − 1)nd−4 .
Thus, in particular, for n = 3 we get
S(K3d ) ≥
4 d
3 .
81
The proof of Theorem 3 will be given in section 4.
2. Basic Definitions
An m-path in a graph is a path containing m vertices, i.e. a path of length m − 1. If P
is an m-path, then we will write m = |P |. A chain C of paths in a graph G is a sequence
(P1 , P2 , . . . , Pm ) of paths in G such that each path in C has at least two vertices, and the
last vertex of Pi is equal to the first vertex of Pi+1 , i = 1, 2, . . . , m − 1. When the number m
of paths in a chain needs to be specified, we shall refer to an m-chain of paths. An m-chain
C = (Pi )m
i=1 of paths will be called closed if the first vertex of P1 is equal to the last vertex
of Pm .
Now we are going to define an important operation that will be used throughout
the paper. Given an m-path P = (gi )m
i=1 in a graph G and an m-chain of paths C =
Pm
(Pi )m
i=1 in a graph H, let P ⊗ C be the
i=1 |Pi | -path in the graph G × H constructed
in the following way.
For each path Pi = (hi1 , hi2 , . . . , hiki ) in C let Pi0 be the path
4
(gi , hi1 ), (gi , hi2 ), . . . , (gi , hiki ) in G × H. Note that for each i ∈ {1, 2, . . . , m − 1} the
0
last vertex of the path Pi0 is adjacent to the first vertex of the path Pi+1
. Let P ⊗ C be the
0
path obtained by joining together (juxtaposing) the paths P10 , P20 , . . . , Pm
. We will say that
P ⊗ C is the path generated by P and C. Note that the path generated by a closed path and
a closed chain of paths is a closed path.
If D is a km-chain of paths in a graph H, then the m-splitting of D is the sequence
(D1 , D2 , . . . , Dm ) of k-chains of paths in H which joined together (juxtaposed) give D. The
above definition of the operation ⊗ can be generalized to the following situation. Let C =
(Pi )m
i=1 be an m-chain of k-paths in a graph G, let D be a km-chain of paths in H, and
let (D1 , D2 , . . . , Dm ) be the m-splitting of D. Note that for each i ∈ {1, 2, . . . , m − 1} the
last vertex of the path Pi ⊗ Di in the graph G × H is equal to the first vertex of the path
Pi+1 ⊗ Di+1 . Set
C ⊗ D = (P1 ⊗ D1 , P2 ⊗ D2 , . . . , Pm ⊗ Dm ).
We will say that C⊗D is the chain of paths generated by C and D. Note that the chain of paths
generated by two closed chains of paths is also a closed chain of paths. It is straightforward
to verify that the operation ⊗ is associative in the following sense.
Property 1. If P is an m-path in a graph G, C is an m-chain of k-paths in a graph H, and
D is a km-chain of paths in a graph K, then
P ⊗C ⊗D =P ⊗ C⊗D .
Let C = (Pi )m
i=1 be a chain of paths in a graph G. We say that C is openly separated if
for i ≤ m − 1 and j = i + 1, Pi and Pj have exactly one vertex in common, and otherwise Pi
and Pj are vertex disjoint. We say that C is closely separated if C is closed, Pi and Pj have
exactly one vertex in common when either i ≤ m − 1 and j = i + 1, or i = 1 and j = m,
5
and otherwise Pi and Pj are vertex disjoint. The following property is a straightforward
consequence of the definitions and its proof is left to the reader as an exercise.
Property 2.
(i ) The path P ⊗ C generated by a path P in a graph G and an openly separated chain C
of open snakes in a graph H is an open snake in the graph G × H.
(ii ) The closed path P ⊗C generated by a closed path P in a graph G and a closely separated
chain C of open snakes in a graph H is a snake in the graph G × H.
If P is a path, then let −P be the path obtained from P by reversing the order of
vertices, and if C = (Pi )m
i=1 is a chain of paths, then let −C = (−Pm , −Pm−1 , . . . , −P1 ) be
the chain of paths obtained from C by reversing the order of paths and reversing every path.
The expression (−1)i X, where X is a path or a chain of paths, will mean X for i even and
−X for i odd. Obviously, the following property is true.
Property 3. If P is an m-path in a graph G and C is an m-chain of paths in a graph H,
then (−P ) ⊗ C = −(P ⊗ −C).
Let C be a km-chain of paths, and let S = C1 , C2 , . . . , Cm be the m-splitting of C. By
the alternate matrix of the splitting S we mean the following (m × k)-matrix A of paths:


A=

C1
−C2
..
.
 Q1
1
  Q12
= .
  .
.
Q21
Q22
..
.
...
...
Qk1
Qk2
..
.
Q1m
Q2m
...
Qkm

(−1)m−1 Cm




where (Q1i , Q2i , . . . , Qki ) is the sequence of paths forming the k-chain (−1)i−1 Ci . The splitting
S will be called openly alternating if for every odd j, 1 ≤ j ≤ m − 1, the paths Qkj and Qkj+1
have exactly one vertex in common, for every even j, 2 ≤ j ≤ m − 1, the paths Q1j and Q1j+1
have exactly one vertex in common, and otherwise the paths Qij and Qi` are vertex disjoint,
i ∈ {1, 2, . . . , k}, j, ` ∈ {1, 2 . . . , m}, j 6= `. Note that the splitting S is openly alternating
6
if for every column of its alternate matrix A the paths in the column are mutually vertex
disjoint except for the shared vertices which are necessary for C to be a chain of paths, i.e. Qk1
and Qk2 have exactly one vertex in common, Q12 and Q13 have exactly one vertex in common,
and so on.
Assume now that the km-chain C is a closed chain of paths and m is even. Then, we
say that the splitting S is closely alternating if for every odd j, 1 ≤ j ≤ m − 1, the paths Qkj
and Qkj+1 have exactly one vertex in common, for every even j, 2 ≤ j ≤ m − 1, the paths
Q1j and Q1j+1 have exactly one vertex in common, the paths Q11 and Q1m have exactly one
vertex in common, and otherwise the paths Qij and Qi` are vertex disjoint, i ∈ {1, 2, . . . , k},
j, ` ∈ {1, 2 . . . , m}, j 6= `. As above, note that the splitting S is closely alternating if for
every column of its alternate matrix A the paths in the column are mutually vertex disjoint
except for the shared vertices which are necessary for C to be a closed chain of paths. The
following property is a straightforward consequence of the definitions.
Property 4. Let P be a k-path in a graph G, and let D be a km-chain of paths in a
graph H.
(i ) If C is the m-chain (P, −P, . . . , (−1)m−1 P ), and the m-splitting of D is openly alternating, then the m-chain C ⊗ D of paths in the graph G × H is openly separated.
(ii ) If m is even, C is the closed m-chain (P, −P, P, −P, . . . , −P ), and the m-splitting of D
is closely alternating, then the closed m-chain C ⊗ D of paths in the graph G × H is
closely separated.
Assume that n ≥ 3 is a fixed odd integer. For every integer d ≥ 1, we are going now to
define the nd -path πnd in Knd , and the closed (n − 1)nd -path γnd+1 in Knd+1 . These paths will
be used in the construction of long snakes.
Let πn1 be the n-path (0, 1, . . . , n − 1) in Kn , and let γn be the closed (n − 1)-path
7
(0, 1, . . . , n − 2) in Kn . If d ≥ 1 and the path πnd in Knd is defined, then let
πnd+1 = πn1 ⊗ πnd , −πnd , πnd , −πnd , . . . , πnd
and
γnd+1 = γn ⊗ πnd , −πnd , πnd , −πnd , . . . , −πnd .
Let H be a graph, d ≥ 1 be an integer, C be an nd -chain of paths in H, and D be an
(n − 1)nd -chain of paths in H. We say that C is openly well distributed if either d = 1 and C
is an openly separated chain of open snakes, or d ≥ 2, every chain Ci in the n-splitting S =
(C1 , C2 , . . . , Cn ) of C is openly well distributed and S is openly alternating. We also say that
D is closely well distributed if every chain Di in the (n − 1)-splitting S 0 = (D1 , D2 , . . . , Dn−1 )
of D is openly well distributed and S 0 is closely alternating. The following property can be
proved by a straightforward induction with respect to d.
Property 5. If C is an openly well distributed nd -chain of paths in a graph H, then the
chain −C is also openly well distributed.
Now we are ready to prove the following important lemma.
Lemma 1. If d ≥ 1 and C is an openly well distributed nd -chain of paths in a graph H,
then the path πnd ⊗ C is an open snake in the graph Knd × H.
Proof. We are going to use induction with respect to d. For d = 1, the lemma is true by
Property 2 (i ). Assume that d ≥ 1, and that C is an openly well distributed nd+1 -chain of
paths in the graph H. Let S = (C1 , C2 , . . . , Cn ) be the n-splitting of C. By the definition,
we have πnd+1 ⊗ C = (πn1 ⊗ D) ⊗ C, where D = (πnd , −πnd , . . . , πnd ). By Property 1, the chain
πnd+1 ⊗ C is equal to πn1 ⊗ (D ⊗ C). We have
D⊗C =
πnd
⊗
C1 , (−πnd )
8
⊗
C2 , . . . , πnd
⊗ Cn
By Property 3,
D ⊗ C = πnd ⊗ C1 , −(πnd ⊗ −C2 ), . . . , πnd ⊗ Cn
Since the chain C is openly well distributed, the chains C1 , C2 , . . . , Cn are also openly well
distributed. By Property 5, the chains C1 , −C2 , . . . , Cn are openly well distributed, so by the
inductive hypothesis the paths πnd ⊗ C1 , πnd ⊗ −C2 , . . . , πnd ⊗ Cn are open snakes in Knd × H.
Hence D ⊗ C is a chain of open snakes in Knd × H. The splitting S is openly alternating, so by
Property 4 (i ), the chain D ⊗ C is openly separated. Hence by Property 2 (i ), πn1 ⊗ (D ⊗ C)
is an open snake in Kn × Knd × H = Knd+1 × H, and the proof is complete.
The following lemma will be used in the proof of Theorem 3.
Lemma 2. If d ≥ 1 and C is a closely well distributed (n − 1)nd -chain of paths in a graph
H, then the path γnd+1 ⊗ C is a snake in the graph Knd+1 × H.
Proof. Assume that d ≥ 1 and that C is a closely well distributed (n − 1)nd -chain of paths
in the graph H. Let S = (C1 , C2 , . . . , Cn−1 ) be the (n − 1)-splitting of C. By the definition
we have γnd+1 ⊗ C = (γn ⊗ D) ⊗ C, where D is the (n − 1)-chain (πnd , −πnd , . . . , −πnd ). By
Property 1, the chain γnd+1 ⊗ C is equal to γn ⊗ (D ⊗ C). We have
D⊗C =
πnd
⊗
C1 , (−πnd )
⊗
C2 , . . . , (−πnd )
⊗ Cn−1
By Property 3,
D ⊗ C = πnd ⊗ C1 , −(πnd ⊗ −C2 ), . . . , −(πnd ⊗ −Cn−1 )
By Property 5 and Lemma 1, D ⊗C is a chain of open snakes in Knd . The splitting S is closely
alternating so by Property 4 (ii ), the chain D ⊗ C is closely separated. Hence by Property 2
(ii ), γn ⊗ (D ⊗ C) is a snake in Kn × Knd × H = Knd+1 × H, and the proof is complete.
9
3. Construction of Well Distributed Chains
Let C be a chain of paths in a graph G. We say that C joins u1 to u2 if u1 is the first vertex of
the first path of C and u2 is the last vertex of the last path. If v is the first or the last vertex
of a path P , then let P − v be the path obtained by removing v from P . Let C = (Pi )m
i=1 be a
chain of paths joining u1 to u2 , and D = (Qi )m
i=1 be a chain of paths joining v1 to v2 . We say
that C and D are parallel if for every i, 1 ≤ i ≤ m, the paths Pi and Qi are vertex disjoint.
We also say that C and D are internally parallel if the following conditions are satisfied:
(i ) the paths P1 − u1 and Q1 are vertex disjoint,
(ii ) the paths P1 and Q1 − v1 are vertex disjoint,
(iii ) the paths Pm − u2 and Qm are vertex disjoint,
(iv ) the paths Pm and Qm − v2 are vertex disjoint, and
(v ) for every i, 1 < i < m, the paths Pi and Qi are vertex disjoint.
Let n ≥ 2 be an integer and let H be a graph. An n-net in H is a pair (U, M), where
U = {u1 , u2 , . . . , un+3 } is a set of n + 3 vertices of H and
n
o
i
M = Ns,t : i ∈ {0, 1, . . . , n − 1}; s, t ∈ {1, 2, . . . , n + 3}; s 6= t
i
i
=
joins us to ut , Ns,t
is a set of openly separated chains of open snakes in H such that Ns,t
j
i
i
are internally parallel, for any i, j ∈ {0, 1, . . . , n − 1},
−Nt,s
, and if i 6= j then Ns,t
and Nv,w
and s, t, v, w ∈ {1, 2, . . . , n + 3}, s 6= t, v 6= w.
Assume now that n ≥ 3 is a fixed odd integer. Let Xn be the following n×(n+3)-matrix.

5
4
7
6
9
..
.
6
5
8
7
10
..
.
3
6
5
8
7
..
.
4
7
6
9
8
..
.






Xn = 



n + 2 n + 3
n
n+1

n + 1 n + 2 n + 3
1
1
2
n+2 n+3
7
8
9
10
11
..
.
1
2
3
10

8
9 ... n + 3
1
2
9 10 . . .
1
2
3 

10 11 . . .
2
3
4 

11 12 . . .
3
4
5 

12 13 . . .
4
5
6 
..
..
..
..
.. 
.
.
.
.
. 

2
3 ... n − 3 n − 2 n − 1

3
4 ... n − 2 n − 1
n 
4
5 ... n − 1
n
n+1
Let Yn be the following (n − 1) × (n + 3)-matrix.

5
4
7
..
.
6
5
8
..
.
3
6
5
..
.
4
7
6
..
.
7
8
9
..
.




Yn = 

n − 1
n
n+1 n+2 n+3

n + 2 n + 3
n
n+1
1
n+1 n+2
4
1
2
8 9 10
9 10 11
10 11 12
..
..
..
.
.
.
1
2
3
2
3
4
3
5
6
...
...
...
n+3
1
2
..
.
1
2
3
..
.
2
3
4
..
.







... n − 4 n − 3 n − 2

... n − 3 n − 2 n − 1
... n − 1
n
n+3
The matrix Xn can be obtained by taking as rows the n consecutive cyclic permutations of
the sequence (3, 4, . . . , n + 3, 1, 2), and then for each odd row exchanging the entries at the
positions 1 and 3 and exchanging the entries at the positions 2 and 4. The matrix Yn can be
obtained from Xn by removing the last row and modifying the n − 1 row.
From now on, to the end of this section, let us assume that we are given a graph H, and
an n-net (U, M) in H. Let C be an nd -chain of paths in H, d ≥ 1. If every n-chain in the
nd−1 -splitting of C belongs to M, then we say that C is M-built. For each i = 1, 2, . . . , n, let
ϕi : M → M be defined by
j
mod n
ϕi (Nv,w
) = Nxj+i−1
,
i,v ,xi,w
where the bottom indices are taken from the matrix Xn = (xp,r ). Analogously, for each
i = 1, 2, . . . , n − 1, let ψi : M → M be defined by
j
mod n
ψi (Nv,w
) = Nyj+i−1
,
i,v ,yi,w
where the bottom indices are taken from the matrix Yn = (yp,r ). If C is any M-built nd chain, then ϕi (C) and ψi (C) are the nd -chains obtained by applying ϕi and ψi , respectively,
to each of the n-chains in the nd−1 -splitting of C.
d−1
d−1
n
Let D1 and D2 be M-built nd -chains with nd−1 -splittings (Nsikk,tk )k=1
and (Nvjkk,wk )nk=1
respectively. If ik = jk for 1 ≤ k ≤ nd−1 , sk = vk for 1 < k ≤ nd−1 , and tk = wk for
1 ≤ k < nd−1 , then we say that D1 and D2 are internally M-compatible. If ik 6= jk for
1 ≤ k ≤ nd−1 , sk 6= vk for 1 < k ≤ nd−1 , and tk 6= wk for 1 ≤ k < nd−1 , then we say that
11
D1 and D2 are internally M-parallel. If ik 6= jk for 1 ≤ k ≤ nd−1 , sk 6= vk for 1 ≤ k ≤ nd−1 ,
and tk 6= wk for 1 ≤ k ≤ nd−1 , then we say that D1 and D2 are M-parallel.
Let C be an M-built nd+1 -chain, let S = (C1 , C2 , . . . , Cn ) be the n-splitting of C, and let

C1
−C2
C3
..
.



A=


 −C
n−1








Cn
be the alternate matrix of S. We say that C is M-well distributed if either d = 0, or d > 0
and the following conditions are satisfied.
(i ) For every i, 1 ≤ i ≤ n, the nd -chain Ci is M-well distributed.
(ii ) Any two consecutive rows of A form a pair of internally M-parallel chains.
(iii ) Any two nonconsecutive rows of A form a pair of M-parallel chains.
Since the n-splitting of an M-well distributed chain is openly alternating, the following
property can be proved by induction with respect to d.
Property 6. If C is an M-well distributed nd -chain, then C is an openly well distributed
chain.
Since no integer appears twice in any row of Xn , no integer appears twice in any row
i
i
of Yn , and Ns,t
= −Nt,s
, a straightforward induction with respect to d can be used to prove
the following property.
Property 7. If C is an M-well distributed nd -chain, then for any i ∈ {1, 2, . . . , n} and any
j ∈ {1, 2, . . . , n − 1} the chains ϕi (C), ψj (C), and −C are also M-well distributed.
Now we are ready to prove the following important lemma.
d
d
d
Lemma 3. For any d ≥ 1 there exist four internally M-compatible nd -chains C3,1
, C3,2
, C4,1
12
d
d
and C4,2
, such that Cs,t
is M-well distributed and joins us to ut , for any s ∈ {3, 4}, t ∈ {1, 2}.
1
0
Proof. We shall use induction with respect to d. If d = 1, then let Cs,t
= Ns,t
, for each
1
1
1
1
, C4,1
and C4,2
have the
s ∈ {3, 4}, and each t ∈ {1, 2}. Clearly, the n-chains C3,1
, C3,2
required properties.
d
d
d
d
Suppose now that d ≥ 1 and the nd -chains C3,1
, C3,2
, C4,1
and C4,2
satisfy the specified
d+1
conditions. Given s ∈ {3, 4} and t ∈ {1, 2}, let Cs,t
be the nd+1 -chain with the n-splitting
d
d
d
S = (C1 , C2 , . . . , Cn ) such that C1 = ϕ1 (Cs,1
), Ci = ϕi (C4,1
) for odd i, Ci = −ϕi (C3,2
) for even
d
).
i, 1 < i < n, and Cn = ϕn (C4,t
d
d
d
d
are internally M-compatible, there are
, C4,1
, and C4,2
, C3,2
Let m = nd−1 . Since C3,1
i1 , i2 , . . . , im ∈ {0, 1, . . . , n − 1} and z1 , z2 , . . . , zm−1 ∈ {1, 2, . . . , n + 3} such that for any
v ∈ {3, 4} and w ∈ {1, 2}, the sequence
i1
im
m−1
(Nv,z
, Nzi12,z2 , Nzi23,z3 , . . . , Nzim−2
,zm−1 , Nzm−1 ,w )
1
d
. Thus, if A is the alternate matrix of the splitting S, then we have:
is the m-splitting of Cv,w
  ϕ1 (N i1 )
d
ϕ1 (Nzi12,z2 )
s,z1
ϕ
(C
)
1
s,1
C1
i1
d

  ϕ2 (N3,z
)
ϕ2 (Nzi12,z2 )
1
 −C2   ϕ2 (C3,2 )  

 
  ϕ3 (N i1 )
d
ϕ3 (Nzi12,z2 )
4,z1
 C3   ϕ3 (C4,1 )  


 

i1
d
)  
)
ϕ4 (Nzi12,z2 )
ϕ4 (N3,z
 −C4   ϕ4 (C3,2

1
=

A=
=
d
i1
 C5   ϕ5 (C4,1
)  
ϕ5 (N4,z
)
ϕ5 (Nzi12,z2 )
1
 .  
 

 .  
 
..
..
..
 .  
 
.
.
.

 −C
 

d
i1
 ϕn−1 (C3,2
n−1
)   ϕn−1 (N3,z
) ϕn−1 (Nzi12,z2 )
1
Cn
d
i1
ϕn (C4,t )
ϕn (N4,z
)
ϕn (Nzi12,z2 )
1



...
...
...
...
...
...
...

m
ϕ1 (Nzim−1
,1 )
m

ϕ2 (Nzim−1
,2 ) 

m
ϕ3 (Nzim−1
,1 ) 

m

ϕ4 (Nzim−1
)
,2


m
ϕ5 (Nzim−1
)

,1

..


.

im
ϕn−1 (Nzm−1 ,2 ) 
m
ϕn (Nzim−1
,t )
d+1
d+1
We will prove now that Cs,t
satisfies the required conditions. To verify that Cs,t
is
a chain of paths, note that the entries of the matrix Xn = (xj,k ) satisfy xi,1 = xi+1,2 for i
d+1
odd, and xi,3 = xi+1,4 for i even, 1 ≤ i ≤ n − 1. Since x1,s = s and xn,t = t, the chain Cs,t
d
joins us to ut . Since the chains Cv,w
are M-well distributed, Property 7 implies that the
chains C1 , C2 , . . . , Cn are also M-well distributed. Since the integers 3 and 4 do not appear
in the 3rd or 4th column of the matrix Xn except in the first row, the integers 1 and 2 do
13
not appear in the 1st or 2nd column of Xn except in the last row, and no integer appears
twice in any column of Xn , it follows from the definition of the functions ϕi that any two
consecutive rows of A are internally M-parallel chains and any two nonconsecutive rows of A
d+1
are M-parallel chains. Hence the chain Cs,t
is M-well distributed. Since it is clear that the
d+1
d+1
d+1
d+1
chains C3,1
, C3,2
, C4,1
and C4,2
are internally M-compatible, the proof is complete.
The following lemma will be used in the proof of Theorem 3.
Lemma 4. For any d ≥ 1 there exists a closely well distributed M-built (n − 1)nd -chain.
Proof. Let d ≥ 1. Let Dnd+1 be the closed (n − 1)nd -chain with the (n − 1)-splitting S =
d
d
(C1 , C2 , . . . , Cn−1 ) such that Ci = ψi (C4,1
) for odd i, and Ci = −ψi (C3,2
) for even i, 1 ≤ i ≤
n − 1. Thus, if A is the alternate matrix of the splitting S, and m = nd−1 , then we have:

m
  ψ1 (N i1 )
d
ψ1 (Nzi12,z2 )
...
ψ1 (Nzim−1
)

  ψ1 (C4,1
)
4,z
,1
1
C1
i1
m

d
ψ2 (Nzi12,z2 )
...
ψ2 (Nzim−1
ψ2 (C3,2
)  
,2 ) 
 −C2  
 ψ2 (N3,z1 )



 


i1
d
m
) 
)
ψ3 (Nzi12,z2 )
...
ψ3 (Nzim−1
 C3   ψ3 (C4,1
 ψ3 (N4,z
,1 ) 

1


A=
=
..
 ..  = 


..
..
..

 .  


.
.
.
.
 

 C
 


d
i
i
i2
1
m
 ψn−2 (N4,z

n−2
ψn−2 (C4,1 )
)
ψ
(N
)
.
.
.
ψ
(N
)
n−2
n−2
z
,z
z
,1
1 2
1
m−1
d
−Cn−1
i1
m
ψn−1 (C3,2 )
ψn−1 (N3,z
) ψn−1 (Nzi12,z2 ) . . . ψn−1 (Nzim−1
,2 )
1
where, for some i1 , i2 , . . . , im ∈ {0, 1, . . . , n − 1}, and z1 , z2 , . . . , zm−1 ∈ {1, 2, . . . , n + 3}, the
sequence
i1
im
m−1
(N4,z
, Nzi12,z2 , Nzi23,z3 , . . . , Nzim−2
,zm−1 , Nzm−1 ,1 ),
1
d
is the m-splitting of the chain C4,1
, and the sequence
im
i1
m−1
(N3,z
, Nzi12,z2 , Nzi23,z3 , . . . , Nzim−2
,zm−1 , Nzm−1 ,2 ).
1
d
is the splitting of the chain C3,2
,.
Since the entries of the matrix Yn = (yj,k ) satisfy yi,1 = yi+1,2 for i odd, yi,3 = yi+1,4
for i even, 1 ≤ i ≤ n − 2, and yn−1,3 = y1,4 , it follows that the sequence Dnd+1 of paths is a
d
closed chain of paths. Since the chains Cv,w
are M-well distributed, Property 7 implies that
14
the chains C1 , C2 , . . . , Cn−1 are also M-well distributed. By Property 6, C1 , C2 , . . . , Cn−1 are
openly well distributed. Since no integer appears twice in any columns of Yn , it follows that
any two consecutive rows of A are internally M-parallel chains and any two nonconsecutive
rows of A are M-parallel chains, hence the splitting S is closely alternating. Thus, Dnd+1 is
a closely well distributed M-built chain, and the proof is complete.
4. Construction of an n-Net in Kn3
Let n ≥ 3 be a fixed odd integer. The vertices of the graph Kn3 are 3-tuples of digits
from the set {0, 1, . . . , n − 1}. Given a vertex v = (a1 , a2 , a3 ) of Kn3 , we say that ai appears
at the i-th position of v, i = 1, 2, 3. If a ∈ {0, 1, . . . , n − 1}, then let a = a +
n−1
2
mod n, and
a = a + n+1
2 mod n. If a1 , a2 ∈ {0, 1, . . . , n − 1}, then let [a1 , a2 ] = {a1 , a1 + 1, a1 + 2, . . . , a2 },
where the addition is taken modulo n. Thus, for example, if n = 7 then [5, 1] = {5, 6, 0, 1}.
We are going now to define the n-net (Un , Mn ) in Kn3 . Let
o
n
n+1
Un = {u1 , u2 , . . . , un+3 } = (a1 , a2 , a3 ) : a1 ∈ {0, 1, . . . ,
}, {a2 , a3 } = {a1 , a1 } .
2
To construct the chains in the set Mn we need to introduce some definitions. Suppose that
v = (a1 , a2 , a3 ) is a vertex from the set Un , and let us assume that the digit b 6∈ {a1 , a2 , a3 }.
The adjunct of the digit b in v is the digit a1 if b ∈ [a1 , a1 ], and the digit a1 otherwise. Let
η : {1, 2, 3} → {1, 2, 3} be the function defined by η(i) = i + 1 if 1 ≤ i < 3, and η(3) = 1. For
any i ∈ {0, 1, . . . , n − 1} and s ∈ {1, 2, . . . , n + 3}, let Qis be the path starting with us defined
as follows. If the digit i appears at the j-th position in us , then let Qis = (us , u0s ), where
u0s is obtained from us by replacing the digit at the η(j)-th position with the digit i. If the
digit i does not appear in us , then let the adjunct of i appear at the j-th position in us . Let
Qis = (us , u0s , u00s ), where u0s is obtained from us by replacing the digit at the j-th position
with the digit i, and let u00s be obtained from u0s by replacing the digit at the η(j)-th position
with the digit i. For example, if n = 5 and the sequence u1 , u2 , . . . , u8 of vertices in U5 is
15
023, 032, 134, 143, 240, 204, 301, 310, where we write a1 a2 a3 instead of (a1 , a2 , a3 ), then the
path Qis is the path in the s-th row and the (i + 1)-st column of the following matrix:
(023, 003)
 (032, 002)

 (134, 130, 030)

 (143, 103, 100)

 (240, 040)

 (204, 200)

(301, 300)
(310, 010)

(023, 013, 011)
(032, 031, 131)
(134, 114)
(143, 113)
(240, 241, 141)
(204, 214, 211)
(301, 101)
(310, 311)
(023, 022)
(023, 323)
(032, 232)
(032, 033)
(134, 124, 122)
(134, 133)
(143, 142, 242)
(143, 343)
(240, 220)
(240, 230, 233)
(204, 224)
(204, 203, 303)
(301, 302, 202)
(301, 331)
(310, 320, 322)
(310, 330)

(023, 024, 424)
(032, 042, 044) 

(134, 434) 

(143, 144) 

(240, 244) 

(204, 404) 

(301, 341, 344)
(310, 314, 414)
It can be verified that the following property is true.
Property 8. If i, j ∈ {0, 1, . . . , n − 1}, i 6= j, and s, t ∈ {1, 2, . . . , n + 3}, s 6= t, then the
paths Qis and Qjs have only the first vertex in common, and the paths Qis and Qjt are vertex
disjoint.
Now we are ready to define the set
n
o
i
Mn = Ns,t
: i ∈ {0, 1, . . . , n − 1}; s, t ∈ {1, 2, . . . , n + 3}; s 6= t
of n-chains of paths in Kn3 . Given i ∈ {0, 1, . . . , n − 1}, and s, t ∈ {1, 2, . . . , n + 3} such that
s < t let
i
Ns,t
= (Qis , P1 , P2 , . . . , Pn−2 , −Qit ),
and
i
Nt,s
= (Qit , −Pn−2 , −Pn−3 , . . . , −P1 , −Qis ),
where P1 , P2 , . . . , Pn−2 are defined as follows. Let vs , vt be the last vertices of the paths Qis
and Qit respectively, let js , jt ∈ {1, 2, 3} be such that i does not appear neither at the js -th
position of vs , nor at the jt -th position of vt , and let as , at be the digits at the position js
of vs and the position jt of vt respectively. Note that for v ∈ {vs , vt }, the digit i appears
at exactly two positions of v, hence js and jt are well defined and i 6∈ {as , at }. Now let us
consider two cases.
16
If js = jt , then let b2 , b3 , . . . , bn−2 be a sequence of different digits from the set
{0, 1, . . . , n − 1} \ {as , at , i}, let b1 = as , bn−1 = at , and let Pj be the path (wj , wj+1 )
in Kn3 , j = 1, 2, . . . , n − 2, where wk is obtained from vs by replacing the digit at the i-th
position with the digit bk , k = 1, 2, . . . , n − 1. If js 6= jt , then let m =
n−1
2 ,
let b1 = as ,
let b2 , b3 , . . . , bm be a sequence of different digits from the set {0, 1, . . . , n − 1} \ {as , i}, let
bm+1 , bm+2 , . . . , b2m−1 be a sequence of different digits from the set {0, 1, . . . , n − 1} \ {at , i},
and let b2m = at . For j ∈ {1, 2, . . . , 2m − 1} \ {m} let Pj be the path (wj , wj+1 ) in Kn3 , and
let Pm = wm , (i, i, i), wm+1 , where wk is obtained from vs by replacing the digit at the
js -th position with the digit bk , for k = 1, 2, . . . , m, and wk is obtained from vt by replacing
the digit at the jt -th position with the digit bk , for k = m + 1, m + 2, . . . , 2m.
Assumming that in the construction of the paths P1 , P2 , . . . , Pn−2 , we choose the sequences of digits to be increasing and containing as small digits as possible, in the case
n = 5, with u1 , u2 , . . . , u8 as above, we have for example:
0
N1,2 = (023, 003), (001, 001), (001, 004), (004, 002), (002, 032)
0
N1,3 = (203, 003), (003, 001), (001, 000, 010), (010, 030), (030, 130, 134)
0
N3,4 = (134, 130, 030), (030, 010), (010, 000, 200), (200, 100), (100, 103, 143) .
The following lemma will be used in the proof of Theorem 3.
Lemma 5. The pair (Un , Mn ) is an n-net in Kn3 .
i
Proof. Clearly, it follows from the construction and Property 8 that Ns,t
is an openly sepai
i
rated chain of open snakes joining us to ut , and Ns,t
= −Nt,s
, for any i ∈ {0, 1, . . . , n − 1}
i
except the first and the
and s, t ∈ {1, 2, . . . , n + 3}, s 6= t. Note that for every path P of Ns,t
last, the digit i appears at least at two positions in each vertex of P . From the above fact and
j
i
are internally parallel, for any i, j ∈ {0, 1, . . . , n − 1}
Property 8 it follows that Ns,t
and Nv,w
and s, t, v, w ∈ {1, 2, . . . , n+3}, such that i 6= j and v 6= w, hence the proof is complete.
Now, we are ready to prove Theorem 3.
17
Proof of Theorem 3. We have Knd = Knd−3 × Kn3 and d − 3 ≥ 2. Let (Un , Mn ) be the n-net
in Kn3 defined in this section (see Lemma 5). By Lemma 4, there is a closely well distributed
Mn -built (n − 1)nd−4 -chain Dnd−3 . By Lemma 2, the closed path
P = γnd−3 ⊗ Dnd−3
is a snake in Knd . Since
d−3 γn = (n − 1)nd−4
and each path in Dnd−3 has length at least 2, we have
S(Knd ) ≥ |P | ≥ 2(n − 1)dn−4 ,
and the proof is complete.
5. Concluding remarks
The construction of long snakes in Knd presented in this paper can, after minor modifications,
be used in the case when n is even, and thus we can use it to prove Theorem 1. However,
the value of the constant cn obtained in such a proof is not as good as when (1) and (2)
are applied (see Introduction). It follows from (2) and (3) that there is a constant c > 0
independent of n such that for any n ≡ 0 mod 4, and any d ≥ 2
S(Knd ) ≥ cnd−1 .
(4)
By the remark in [5], the constant c can be taken in such a way that (4) holds for all even n.
A natural question arises.
Question 1. Is there a constant c > 0 such that
S(Knd ) ≥ cnd−1 ,
for every n ≥ 2 and d ≥ 2.
We would like to ask a stronger question.
18
Question 2. Is there a constant c > 0 and an integer d0 such that for every n ≥ 2 there
is an n-net (Und0 , Mdn0 ) in the graph Knd0 , where every chain C = (Pi )ni=1 in the set Mdn0
satisfies
n
X
|Pi | ≥ cnd0 .
i=1
It follows from the proof of Theorem 3 that the positive answer to Question 2 implies
that the answer to Question 1 is also positive.
The best known upper bound on S(K2d ) has been given by Snevily [7]
S(K2d )
d−1
≤2
2d−1
−
.
20d − 41
In the general case, Abbott and Katchalski [3] proved that
S(Knd )
≤
1
1+
d−1
nd−1 .
References
[1] H.L. Abbott, P.F. Dierker, Snakes in powers of complete graphs, SIAM J. Appl.
Math. 32 (1977), 347-355.
[2] H.L. Abbott, M. Katchalski, On the snake in the box problem, J. Comb. Theory Ser.
B 45 (1988), 13–24.
[3] H.L. Abbott, M. Katchalski, Snakes and pseudo snakes in powers of complete graphs,
Discrete Math. 68 (1988), 1–8.
[4] H.L. Abbott, M. Katchalski, On the construction of snake in the box codes, Utilitas
Mathematica 40 (1991), 97–116.
[5] H.L. Abbott, M. Katchalski, Further results on snakes in powers of complete graphs,
Discrete Math. 91 (1991), 111–120.
[6] A.A. Evdokimov, The maximal length of a chain in the unit n-dimensional cube. Mat.
Zametki 6 (1969), 309–319. English translation in Math. Notes 6 (1969), 642–648.
[7] H.S. Snevily, The snake-in-the-box problem: a new upper bound, Discrete Math., to
appear.
[8] J. Wojciechowski, A new lower bound for snake-in-the-box codes, Combinatorica 9
(1989), 91–99.
19
[9] J. Wojciechowski, Covering the hypercube with a bounded number of disjoint snakes,
Combinatorica, to appear.
Department of Mathematics, West Virginia University,
PO BOX 6310, Morgantown, WV 26506-6310, USA
E-mail:
UN020243@VAXA.WVNET.EDU
JERZY@MATH.WVU.EDU
20