BULL. AUSTRAL. MATH. SOC.
VOL. 29 ( 1 9 8 4 ) ,
05C38,
O5C45
I-1 I .
CYCLABILITY OF /--REGULAR r-CONNECTED GRAPHS
W.D. McCUAIG AND M. ROSENFELD
For each value of
r-regular,
than
r > U,
r
even, we construct infinitely many
reconnected graphs whose cyclability i s not greater
6r - k
if
r = 0 (mod h)
1.
and
8r - 5
if
r = 2 (mod U) .
Introduction
We use the terminology and notation in Bondy and Murty [ I ] .
ayalability
of a graph
vertices of
G l i e on a common cycle.
studied by Chvatal [2].
that any
k
G i s the largest integer
We denote by
vertices in an
a common cycle.
k
such that any k
This notion was introduced and
f{r)
r-regular,
the largest integer
r-connected graph
Various authors investigated the function
r = 3 , the Petersen graph shows that
and Thomassen [4], proved that
The
f(3) 5 9 .
/(3) = 9
k
such
(r > 3)
f(r)
.
l i e on
For
Ho I ton, McKay, Plummer
and constructed an infinite
family of cubic graphs (based on the Petersen graph) with cyclability
Holton [ 3 ] , proved that
f(r)
Kelmans and Lomonosov [7].)
Hamiltonian
r-regular,
>r +h .
(This result was also obtained by
Meredith [S], described a construction of non-
r-connected graphs for a l l
that Nash-Williams' conjecture, that all
are Hamiltonian is false).
9 •
r > U (thus showing
U-regular,
U-connected graphs
Meredith's construction is based on the
Petersen graph, and yields the upper bound
f{r)
5 lOr - 11 .
I t i s not
obvious how one can obtain from Meredith's construction, for each value of
Received 29 August 1983. Support from NSERC and SFU Department of
Mathematics i s gratefully acknowledged.
Copyright Clearance Centre, Inc. Serial-fee code:
$A2.00 + 0.00.
OOOU-9T27/8U
I
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2
W.D. McCuaig and M. Rosenfeld
i» , infinitely many graphs with cyclabllity not greater than
lOr - 11 .
In this paper we modify Meredith's construction, and obtain, for each
r , infinitely many
6r - k
2>-regular
[r 5 o (mod k))
and
reconnected graphs with cyclabilities
8r - 5
us to construct non-Hamiltonian,
(such a graph on
8k
r-regular,
v e r t i c e s , for
2.
(r = 2 (mod k))
.
Our method enables
reconnected bipartite graphs,
r = k
is described.)
Meredith's construction
Meredith's construction of
r-regular
reconnected non-Hamiltonian
graphs is based on the following two steps:
(i)
in a graph
G , add edges parallel to existing edges, so
that the resulting multigraph is
r-regular and
r-edge
connected;
(ii)
replace each vertex of the multigraph obtained above by a
copy of
of
K
, connecting the
K
,
r,r-l
with the
r
r
(r-l)-valant vertices
edges incident with the
substituted vertex.
Let
steps.
G
denote the graph obtained from a graph
Meredith proved that
Hamiltonian, i f and only i f
applicable is called
r-good for a l l
J°-regular
G
G is.
r*-good.
r >k .
is
r-regular,
A graph
G , to which step (i) is
I t is easy to obtain, in each of these graphs, a set of
vertices that do not l i e on a common cycle.
[5],
[ 6 ] , in t h e i r study of longest cycles in
graphs, modified Meredith's construction.
Hamiltonian graph
G , that i s
3-connected graphs are
r-good for
reconnected graph
H\{h]
K
Jackson and Parsons
r-regular,
reconnected
They pointed out that any non-
r»-good may be used.
that one i s not restricted to use
such that
is
Since the Petersen graph is not Hamiltonian, a l l
- 10
N{h)
(7
Meredith showed that the Petersen graph is
lOr
r-regular,
reconnected.
r-connected graphs obtained by the above construction are not
Hamiltonian.
cubic
G by the above
r > U.
They proved that a l l
They also point out,
in step ( i i ) .
Actually any
H , for which there is a vertex
h € V(H)
cannot be covered by two disjoint paths with endpoints in
can be used in step ( i i ) .
All these constructions arc based on non-
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r-regular
r-connected
graphs
Hamiltonian graphs, and they do not necessarily yield, infinitely many
r-regular r-connected graphs with cyclability lOr - 11 .
3. Construction of r-regular r-connected graphs
In this section, we describe a modification of Meredith's
construction. This modification allows a much greater flexibility in
choosing the underlying graph (we do not have to start with a nonHamiltonian graph). We assume that r = 0 (mod 2) .
Let
G be an r - r e g u l a r ,
r-connected graph.
i = 1, . . . , r / 2 , be d i s j o i n t edges of
r-connected graph,
G r> Hp = 0 .
be the neighbors of y
F = [G\[elt
G.
Let e. = [a .
Let H
be any
Let N(y) = {y^, ... , yj
. . . , er/2})
(Observe that many graphs
e . }
and ordering
by the
from
» {Hp\{y})
" {{g.,
Vj)
\ 0 = 1, . . . , r}
e
/p} ^
^p '
F can be produced by a fixed choice of
and H , we s t i l l have the freedom of choosing
Let
G be an r-regular,
G by replacing
r-regular
r-connected
, y € V(# r )
y €
N(y) .)
LEMMA 1.
obtained
r-regular,
in H . W e say t h a t the graph
{e-,> •••>
\e , ...,
, g.) ,
r-connected
the
r-conneeted
r / 2 disjoint
graph
H .
Then
edges
graph.
Let
F be
{ e , , . . . , e ,_}
F is an
r-regular,
graph.
Proof.
The proof i s e s s e n t i a l l y s i m i l a r t o the proof of Lemma •*
' in
Rosenfeld [ 9 ] ; we omit the d e t a i l s .
To obtain our construction, we add t o Meredith's construction the
following s t e p :
(iii)
Let
let
G
1-factor
replace
{ e ^ • • •, %/g}
b
y
H
r
•
G and G' be the graphs in Figure 1 (see p. k).
be the multigraph obtained from
If r = km ,
G by replacing the edges of the
A.B. , £ = 0 , 1 , 2 , by 2m parallel edges each and all other
edges by m
parallel edges each.
If r = km + 2 , let G
be the
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W.D.
M c C u a i g and M. R o s e n f e l d
FIGURE 1
multigraph obtained from
by
G' by replacing
A D
Q 0
m p a r a l l e l edges each,
each,
BnBn
by
A C , A CQ, CD
A D
i i
2m parallel edges and A .B.
bjr
m +
^
, and C DQ
Vaxallel
and CD. ,
edges
i = 0, 1 by
2m + 1 parallel edges each.
LEMMA 2.
Proof.
G
C
is
The lemma is proved by showing that any pair of vertices of
are connected by
CASE 1.
(1)
(3)
r-edge-connected.
r
r = 4m .
A.B.
edge-disjoint paths.
The pairs of vertices are:
,
i = 0, 1, 2 ,
,
i
* 3
,
i-y 5 = 0 , 1 , 2 .
A.A.,B.B.\
By symmetry, i t suffices to consider one example from each case.
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r-regular
Case
Edge
r-connected graphs
types of Paths
Number of Paths
1
Vo,.
Vo- ViVo- VaVo
2
V l
VoV Vo¥l' Wl* VzVl
3
Vl
W l V VoVlV W l ' Vl
CASE
2.
r = 4m + 2 .
(1) Aft ,
(2)
A.C.+1
(3)
A.D ,
(U)
V
(5)
B.C. ,
(6)
CiD.+1
(T)
CiD.
(8)
ASU1
» >m
m> m m m
' '
The pairs of vertices a r e :
(9) V l '
,
(10)
(11) A.D.
,
l
2m m
,
i = 0 , 1 (mod 2) ,
(12) S.
(13) B.D
,
,
,
s
(Ik)
B.D.+1
(15)
CQCI ,
,
(16) DQD± .
Again, by symmetry, it suffices to consider one example from each case.
Case
1
Edge
Types of Paths
Vo Vo- VoWiV ViWi s o
Number of Paths
2m+1 m m
' -
Vo c o s o
> m>
3
m
>
Vo Vo'
' Wo°o
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W.D.
Case
Edge
k
Vi
5
McCuaig and M. K o s e n f e l d
Types of Paths
s
oV V o W i - VoWA
V0V1V W1V1
V o Vo^o^o' W i c o ' Voci°oco
ViWc Vo
Number of Paths
2/77, m , m
1, 1
m + 1 , m, m
m, 1
m + 1 , tn, rn
m-1, 1, 1
W i W o - Woflo
2m+l , m , m , 1
2m, m, m
Vi VA-
1, 1
Vi
2m, m , m
WiV WoV
1, 1
10
11
Vo W o ' VoViV W i W o
' Voco
w + 1 , m, m
^o°i V o W r
m+1, m - 1 , 1
m, 1
w, m , 1
12
B QCI
B ^ ^ ^ . VA C O C 1 C 1'
WO
13
C
I-
B c D c
VOC1
o o i i' V o W i B i c i
Vo W o - VoVo- Vo°o
m+1,
m-1, m
m, 1 , 1
m + 1 , m, 1
m, m
m+1, m - 1 , m
m, 1 , 1
m, m , m
m, 1 , 1
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r-regular
Case
16
Edge
r-connected graphs
types of Paths
Number of Paths
Vi W r W r VoW WA
By the terminology of the previous section, G is r-good for a l l
v = 0 (mod 4) and G' is r-good for a l l r = 2 (mod 4) .
THEOREM 1. For r i 4 ,
r-regular
r = 0 (mod 4) there are infinitely
r-connected graphs with cyclability
not greater than
many
6r - 4 .
Proof. For each value of r we f i r s t construct an infinite family of
r-regular, r-connected graphs as follows: l e t G be the multigraph
described in Lemma 2 , Case 1. Let G' be the graph obtained from G
after applying step ( i i ) of Meredith's construction.
theorem,
G' is an r-regular
obtained from
with
A.B. , £ = 0 , 1 , 2 , in G , by an arbitrary
r-connected graph H .
54 vertices, obtained from
indicated edges.)
Let
By Meredith's
Let G" be a graph
G' by replacing the r / 2 disjoint edges in G' ,
corresponding to the edges
r-regular,
r-connected graph.
(Figure 2, p. 8, shows a graph
G! by using
K
Gf ,
for replacing the
By Lemma 1, G* i s r-regular and r-connected.
5 c y(G*)
contain the r - 1 vertices of the smaller color
class of each copy of the six K
's
used in step ( i i ) and also contain
one vertex from each of the three replacement graphs H
CLAIM. No cycle in G* contains
does exist.
Since
S
. Assume that such a cycle
C contains a l l the vertices in {S , . . . , S
C ,
} (Figure
3» P- 9 ) , i t must contain a l l vertices of the corresponding K
r—l ,r
Therefore, C can use exactly two edges from the set
{e , ..., e,, f , ..., fy) . Similarly,
C uses exactly two edges from
the set [e^, ...,«£,/.[, ..., f[) .
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W.D. McCuaig and M. Rosenfeld
FIGURE
Since a vertex of
two edges from
H
2
is contained in
C , C
{e , ..., e, , e', ..., e'\ . If
{e , •••> £,} > it cannot contain any edge from
C
has either zero or two edges from
and connot contain any
K
{/', ...,/'} . But then
1
has two edges from
{f , ..., f1} . Therefore
{e', ..., ej}
1
edge from the set
C
must contain at least
C
cannot contain any
L
vertices outside the configuration in Figure 3have to contain one edge from each of
It follows that
will
f e', ..., ej} ,
{e , ..., e,} ,
X
C
rt
-L
K,
{f, ,...,/,} and \f< ..., /'} .
Contraction of each copy of
K ,
to a single vertex would yield a
J°-l ,2°
cycle in
G
that contains the three edges
easy to see that such a cycle does not exist.
Since
>
and
This proves our claim.
\S \ = 6(r>-l) + 3 , the cyclabilxty of each graph
G*
thus obtained
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r-regular
r-connected graphs
r-1
G*
r
G(G')
FIGURE
is at most
3
6r - h as claimed.
THEOREM
2. For r > 6 , r = 2 (mod U)
there are infinitely many
r-regular r-connected graphs with ayclability not greater than &r - 5 •
Proof.
Let G
be the multigraph described in Lemma 2, case 2. Let
G' be obtained from
to it, and let
G
by applying step (ii) of Meredith's construction
G* be obtained from
G' by replacing the
edges of G' , corresponding to the edges
G' by arbitrary copies of r-regular,
set
S c Vic*) , containing the
class of each
B , CD
r-connected graphs
8(r-l)
r/2 disjoint
, and C' D in
H
. For the
vertices of the smaller color
K
and a single vertex from each of the four graphs
r—l ,r
H
, an identical argument to the proof of Theorem 1, shows that any cycle
in
G* containing
A B , A B , CD
S
, and
, will yield a cycle in G' containing the edges
CD
. Since such a cycle does not exist, S
not contained in a cycle. Hence the cyclability of G* is at most
is
8r - 5
as claimed.
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10
W . D . McCuaig and M. Rosenfeld
4.
Concluding remarks
The modification of Meredith's construction enables us to construct
many
r-regular,
r-connected graphs with prescribed properties.
idea is to start with an
contained in a cycle.
The basic
r-good graph in which some edges are not
In Figure k a li-connected,
Hamiltonian bipartite graph with
l*-regular non-
8k vertices, is described.
This graph,
based on the Mobius ladder, uses the fact that no cycle of the Mobius itladder uses the four "vertical edges".
We believe that the upper bound for the cyclability of
r-regular
^-connected graphs can be further improved by other choices of graphs. It
seems though that another idea for odd
FIGURE
r
is needed.
4
References
[J]
J.A. Bondy and U.S.R. Murty, Graph theory with applications (American
Elsevier, New York, 1976).
[2]
Vaclav Chvatal, "New directions in Hamiltonian graph theory", New
directions in the theory of graphs, 65-95 (Proc. Third Ann Arbor
Conf. on Graph Theory, University of Michigan, Ann Arbor,
Michigan, 1971. Academic Press, New York, 1973).
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http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S0004972700021213
r-regular
[3]
r-connected graphs
D.A. Holton, "Cycles through specified v e r t i c e s in
II
k-connected
regular graphs", Ars Corribin. 13 (1982), 129-1 **3.
[4]
D.A. Holton, B.D. McKay, M.D. Plummer and C. Thomassen, "A nine point
theorem for 3-connected graphs", Combinatoria 2
[5]
Brad Jackson and T.D. Parsons, "On r - r e g u l a r
Hamiltonian graphs", Bull.
Austral.
(1982), 53-62.
r-connected non-
Math. Soo. 24 (1981),
205-220.
[6]
Brad Jackson and T.D. Parsons, "A shortness exponent for
r-regular
r-connected graphs", J. Graph Theory 6 (1982), 169-176.
[7]
A.K. Kelmans and M.V. Lomonosov, "When m v e r t i c e s i n a
graph cannot be walked round along a simple c y c l e " ,
fc-connected
Discrete
Math. 38 (1982), 317-322.
[S]
6.H.J. Meredith, "Regular
w-valent
n-connected non-Hamiltonian non-
n-edge colourable graphs", J. Combin. Theory Ser. B 14 (1973),
55-60.
[9]
M. Rosenfeld, "Are a l l simple
^-polytopes Hamiltonian?", Israel
J.
Math, ( t o appear).
Department of Mathematics,
Simon Fraser University,
Burnaby,
British Columbia,
Canada
V5AIS6;
Department of Mathematics,
Ben Gurion University of Negev,
Beer Sheva
84120,
Israel
and
Department of Mathematics,
University of Washington,
Seattle,
Washington
98195,
USA.
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