I nternat. J. Math. Math. S ci.
Vol. 6 No. 3 (19831 521-533
521
PATH DECOMPOSITIONS OF CHAINS AND CIRCUITS
E.J. FARRELL
Department of [athematics
The University of the West Indies
St. Augustine, Trinidad
West Indies
(Received June 28, 1982 and in revised form January 25, 1983)
ABSTRACT.
Expressions for the path polynomials (see Farrell [I]) of chains and cir-
cuits are derived.
These polynomials are then used to deduce results about node dis-
joint path decompositions of chains and circuits.
Some results are also given for de-
compositions in which specific paths must be used.
KEY WORDS AND PHRASES. Path polynomial, path decomposition, incorporated graph,
etriced decomposition.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 05C38, 05A15.
i.
INTRODUCTION.
The graphs considered here will be finite, undirected and without loops or multi-
ple edges.
w
Let G he such a graph.
With every path
in G, let us associate the weight
With every spanning subgraph C of G consisting of paths
l,e2,...,k,
let us asso-
clare the monomial
k
w()--
W
il
i
Ehe path polynomial of G is
P(G;w_
Zw(C),
where the summation is taken over all the path covers (spanning
Subgraphs
consisting
of paths only) of G. The vector w_ is called the general weight vector. In this paper
we will assign weights to paths according to the number of nodes which they contain.
herefore, we will have w_w
(Wl,W2,...,Wk,...) and P(G;w_) will be a polynomial in the
If we put w
w, for all i, then the resulting polynomial
i
P(G;w) will be called the simple path polynomial of G. The path polynomial is a spec!l F-polynomial (see [i]).
indeterminates
Wl,W2,
etc.
By path incorporating an edge x in G we will mean that x is distinguished in some
anner and required to belong to every path cover of G that we consider. An incorporated graph will be a graph whose edges are all incorporated.
incorporated subgraph will be called a restricted graph.
A graph containing an
Sometimes it will be conven-
522
E. J. FARRELL
lent to shrink an incorporated subgraph to a "node".
compound node.
We will call such a "node" a
A graph G containing an incorporated subgraph or a compound node i.e.
a restricted graph, will normally be denoted by
G*.
A tree with nodes of valencies i and 2 only will be called a chain.
with n nodes will be denoted by P
n
will mean that a terminal node of P
The chain
By attaching a chain P to a nonempty graph G we
n
is identified with a node of
n
connected component in which G and P
n
G, so as
to form a
are subgraphs with exactly one node in common.
By adding a chain P to a graph G, we will mean that the two terminal nodes of P are
n
n
attached to different nodes of G (N.B. In this case, G must have at least two nodes).
We refer the reader to Harary [2] for definitions of the standard terms used in graph
theory.
Upper limits of summations will be infinity unless otherwise specified.
disjoint path decomposition of G is another name for a path cover of G.
A node
The name is-
land decomposition was used by Goodman and Hedetniemi [3], who derived an efficient
algorithm for finding hc(T)
the minimum number of edges needed to be added to a tree
T in order to make it Hamiltonian.
hc(G) is called the Hamiltonian completion number
of G.
Some results on (G) were obtained by Boesch,
It will be denoted by (G).
and McHugh [4].
Cher
Some results for trees were obtained by Slater [5].
In this paper we will use some of the basic results on path polynomials in order
to derive the path polynomials of chains and circuits.
From these polynomials, we will
deduce results about node disjoint path decompositions of the graphs.
Throughout the
paper "decomposition" will mean "node disjoint path decomposition" and
"cover"
mean "path
2.
will
cover".
BASIC PRELIIINARY RESULTS.
By putting the covers of G into classes according to whether or not they contain
a specified edge, we obtain the following
theorem, called the fundamental theorem for
path polynomials.
Let G" be the
Let G be a graph and e an (unincorporated) edge of G.
THEOREM i.
graph obtained from G by deleting e and G*, the graph obtained from G by path incorporating e.
Then
P(G;w_)
(N.B
P(G’;w_) + P(G*;w_).
we will refer to graphs obtained by deleting edges as G
Throughout this paper
and graphs obtained by incorporating edges as G*, whenever Theorem I is applied.)
The fundamental algorithm for path polynomials consists of repeated applications
of Theorem i, until graphs
Gi
are obtained for which
lemmas imply some useful simplifications
P(Gi;w)_
are known.
to the fundamental algorithm
The following
(also called the
reduction process).
LEMLA i.
(i)
If an incorporated edge creates an incorporated circuit in G*, then
P(G*;w)
0.
(ii) If an incorporated edge creates a node of valency 3 in G*, then
P(G*;w__)
0.
PATH DECOMPOSITIONS OF CHAINS AND CIRCUITS
523
In any practical application of the reduction process, we could incorporate an
edge e, by identifying the nodes at the ends of e as described in Read [6] for chro-
However, we must keep track of the number of nodes in the incor-
matic polynomials.
porated subgraph associated with the new node formed by identification.
Lemma 1 suggests that we can delete from G*, all edges which will finally complete
Also, if two edges have been incorporated at node
circuits with incorporated edges.
x, then all unincorporated edges
at x can be deleted.
It is also given in [i]
The following result is called the Component Theorem.
for general F-polynomials.
THEOREI 2.
It can be easily proved.
Let G be a graph consisting of n components
n
P(G;w)
Then
P(Hi;w).
i=l
3.
HI,H 2,... ,Hn.
PATH POLYNOMIALS OF CHAINS.
The notation G*[n] will be used for a restricted graph containing an incorporated
The following theorem gives a recurrence relation for the path
chain with n edges.
polynomial of chains.
TH EOREM 3.
n
P(Pn;W)-PROOF.
w
i=l
P(Pn_i;w)._
i
Apply the reduction process to P
n
by deleting a terminal edge.
an isolated node and P
will be a graph containing two components
Then
G’
G* will be
n-l"
P *[I].
Apply the process to P *[I] by deleting the terminal edge adjacent to the inn
n
corporated edge. In this case, the graph G’ will be a graph with two components, an
G* will be P *[2]. Continue the reand
2
n
duction process in this manner to subsequent graphs P *(k) (k > 2), until the entire
incorporated edge
having a weight w
Pn-2"
n
chain is incorporated.
The result then follows by adding the contributions of the
graphs G’ formed during the process and the final incorporated graph P *[n-i], having
n
a weight w
n
For brevity, we will write P(n) for P(P ;w), when it would lead to no confusion.
Conventionally, we write P(0)
n
1,2,3,4,5 and 6.
the n
th
row is
I.
n
The following table gives values of P(n) for
Observe that the sum of the coefficients of the polynomial in
2n-i
The following theorem gives a generating function P(P
n
;w,t) for P(n).
THEOREM 4.
P(Pn --;w’t)
[i
PROOF.
P(n)
(w t +
I
w2 t2
+ w 3 t 3 + ...)]
n
Y. w. P(n- i)
i=l
coefficient of t
n
w t
in
s=l
Put w(t)
Y.
s=l
w t
s
s
and
P(t)
.
s
Y. P(k)t
k=O
k=0
k.
P(k)t
k
524
E.J. FARRELL
Table i
Path Polynomials of Chains
P(n)
n
1
w
2
w
I
3
w
I
4
4
Wl +
I
2
+ w2
3
+
+ w3
2WlW 2
3w12w + 2WlW + w22 +
13w2 + 3w12w + 3WlW2 + 2WlW4 + 2w2w3 + w5
I +
w16 + 5w14w + 4w13w + 6w12w22 + 3w12w4 + 6WlW2W + 2WlW
w
6
5
2
w
3
4
2
4w
3
2
3
3
+ w2 +
Then
3
2w2w 4
I P(n) t n
5
2
+ w3 + w6
w(t) P(t)
n=l
---->
P(t)
w(t) P(t).
P(0)
The result therefore follows.
We will now consider chains containing incorporated subchains.
The covers of
these restricted chains will be covers in which a particular path must appear.
When
the incorporated subchain contains r edges and contains a node of valency 1 of the
restricted chain, the graph will be denoted by P
n
*[r], where
n is the number of nodes
in the nodes in the unincorporated subchain (including the node common to both sub-
chains).
The path polynomial of P *[r] is given in the following theorem.
n
THEOREM 5.
n
P(en*[r];w)
PROOF.
Wr+i
i=l
P(n-i).
Apply the reduction process to P *[r] by always deleting the edge which
n
links the restricted edges to the unrestricted ones.
The result then follows.
When the incorporated subchain does not include a node of valency i of the restricted chain, the graph will be denoted by P
*
nl+n 2
[r] where
the number of nodes in the unrestricted subchains.
n
I
and n
2
(nl,n 2 > i)
are
The path polynomial of this graph
is given in the following result.
THEOREM 6.
n
e(en+n2[r ];w_) s=l t=l Wr+s+t-i P(nlPROOF.
subchain
Pnl
s) P(n
2-
t)
(nl,n 2 > 1).
Apply the reduction process to the graph, by deleting the edge of the
two components
which is adjacent to an incorporated edge.
Pnl_l
and
P*n2[r].
GI*
will be
The graph G
P(n_l)+n2[r+l].
I
will consist of
Apply the reduction
PATH DECOMPOSITIONS OF CHAINS AND CIRCUITS
GI*, and to subsequent Go*’Sl by deleting the
Pn.1 which is adjacent to an incorporated edge.
edge of the unrestricted sub-
process to
chain of
of the
G"s
1
and the final restricted graph P
n
P(P **
nl+n 2
n2[nl
*
525
By adding the contributions
-l+r] we get
I
[r];w)
P(Pn[r+s-l];w).
P(nl-s
s=l
The result follows by using Theorem 5.
When the incorporated chain is attached to a node of valency 2 of the chain, the
graph will be denoted by P
*
nl,n 2
[r], where
n
(nl,n2 > i)
I and n 2
are the number of nodes
The result for this graph is gi-
in the subchains separated by the node of valency 3.
ven in the following theorem.
THEORY24 7.
P(en n2[r];w--)
PROOF.
n
e(nl-1
ni_ 1
2
Wr+i P(n2-i) +P(n2-1)
i=l
Wr+j+I
i=l
> I).
P(nl-J-l) (nl 2
Let x be the node of valency 3 and y and z the nodes of the subchains P n
1
respectively,
Pn2
ting the edge xy. G
and
Apply the reduction process by dele-
which are adjacent to x.
will be the graph consisting of components
will consist of components
and
Pn2_l
P(Pn,n 2[r];w)-
Pnl_l
[r+l]
+ P(n 2- I)
P(Pn[r];w)_
P(n I- i)
Pnl_ I
and
Pn[r].
G*
Hence we get
P(Pn_l[r+l];w)._
The result then follows by using Theorem 5.
4.
S I{PLE PATH POLYNOMIALS OF CHAINS.
The factor (i +w) appears quite frequently in the simple path polynomials of
We will therefore replace it by z.
chains and circuits.
P(G;w)
where the
A.’s
l
Aiz
By putting w
are constants.
i=l
will be the coefficient of w in P(G;w), which is
Let P(G;w)
(i)
[
i=l
and
Z Ao
O, we get
(ii)
0, since P(G;w) can
Also, the number of Hamiltonian paths in G (denoted by H(G))
have no constant term.
LEMMA 2.
Let us assume that
i
H(G)
A
i
[
A
[
i
i=l
0
i
i=l
z
i
where z
7 i A
i.
i=l
+ w.
1
Hence we have
Then
A..
1
We can obtain the simple path polynomial of P
for all i, then equating coefficients of t
n.
from Theorem 4 by putting w
n
i
w
However, the result can be more easily
obtained by a straightforward combinatorial argument. A cover of P with cardinality
n
k can be obtained by deleting k-1 edges for any 0 < k < n.
THEOREM 8.
P(Pn;W)
w(l+w)
n-i
z
n
z
n-i
Hence, we have
526
E.J. FARRELL
The following corollaries are immediate from Theorems 5 and 6.
They can also be
independently proved, by recognizing that in the case of the simple path polynomial
of restricted chains, we can "shrink" the incorporated subchains to a node.
COROLLARY 5.1.
For all r,
e(Pn;W)
P(Pn* [r];w)
COROLLARY 6.1.
z
n
z
n-i
For all r,
P(P *.*
nln 2
[r];w)
P(P
nl+n2-1
;w)
z
nl+n2-1
z
nl+n2-2
[r] by direct substitution into
We can obtain the simple path polynomial of P *
n n
I 2
It is better to obtain the result
Theorem 7; however, this will be a bit tedious.
The result is given in the following
independently by using the reduction process.
corollary, for which an independent proof is given.
COROLLARY 7. i.
P(P *
nl,n 2
z
nl+n2-4 (z- i) 2
(z+l).
Apply the reduction process to the graph, just as we did in the proof of
PROOF.
Theorem 7.
[r];w)
G
will consist of components P
P(P *[r];w)
n
and P
nl-i
P(P
2
n2
n
But from Corollary 5.1,
2
;w).
Pn2_l and Pnl_l* [r+l].
G* will consist of components
*[r].
But
P(PnI_I* [r+l];w) P(Pnl_l;W).
Hence we get
P
(Pn ,n 2 [r];w)
P(Pn
i)
1
P(n2)
+ P(n 2
nl-2 )(z n 2
nl-I
z
I) P(n
n2-1
I
i)
+ (z n2-1
z
n2-2 )(z nl-i
z
nl-2
The result then follows after simplification.
k*
Let us denote by
the restricted cha+/-n formed by attaching to a chain
Pn
Pn’
k
[n/2] such that no two are attached to adjacent nodes of Pn
We will also assume that none of the end-nodes of P are used, for if it is, then it
incorporated paths (k <
n
can be treated as an ordinary node, as implied by Corollary 5.1.
polynomial of P
k*
n
The simple path
is given in the following theorem.
THEOREM 9.
P(P
n
k*;w)
PROOF BY INDUCTION ON k.
z
P(P n
(z- l)
k+l
(z+l
)k.
Corollary 7.1 verifies the case for k= 1.
sume that the statement holds for k
r*
n-2k-1
;w)
r.
z
Let us as-
Then
n-2r-1
(z- i)
r+l
(z+l)
r
We can apply the reduction process to P (r+l)* by deleting the edge which is adjacent
n
PATH DECOIIPOSITIONS OF CHAINS AND CIRCUITS
to
527
"last" incorporated path from the end of the restricted chain, such that G would
consist of (i) a restricted chain containing r incorporated paths, and (ii) a restric-
Thus the components of
ted chain with one end-node attached to an incorporated path.
will be P
G
r*
s
and P
t
*
with s+t
Pt*’ P(Pt*;w) P(Pt;w).
end-node of
P(G ;w)
z
z
Since the incorporated path is attached to an
n
s-2r-i
n-2r-2
Hence
(z-l)
(z-l)
r+l
(z+l)
r+2
(z+l)
pr*
The graph G* will contain two components
P(G*;w)
z
z
s-2r-2
n-2r-3
(z- i)
(z-l)
(z+l)
r+2
z
(z+l)
t-i
(z- i)
r
and P
s-i
r+l
r
r
Hence
t"
z
t-i
(z- i)
r
It follows that
p(p
(r+l)*
n
;w)
z
n-2r-2
(z-l)
r+2
r
(z+l) +z
n-2r-3
(z- i)
r+2
(z+l)r.
On simplification, we get
P(Pn
(r+l)*;w)
z
n-2r-3
(z- i)
r+2
(z+l)
r+l
r+l, when it is assumed for k
Hence the statement holds for k
By the Princi-
r.
ple of Induction, it holds for all r.
5.
PATH DECOMPOSITION OF CHAINS.
n kr
Let
(nl kl n2 k2
r
the number of decompositions of the graph G into k
N
1,2,...,r.
i
paths with
NG()
no nodes, for
It
will denote the total number of decompositions of G into paths.
G
NG()_
is clear that
We will denote by
be a partition of n.
is the coefficient of w
klnl
wk2
n
w
kr
n
in
P(G;w).
r
2
It confirms the observation
The following corollary is immediate from Theorem 8.
made about the sum of the coefficients of the polynomials in Table i above.
COROLLARY 8. i.
Npn
2
n-l.
It can be observed from Table i, that all partitions of the integer n appear as
subscripts of w.
[w ki
This suggests the following theorem
(k times)].
WlW I
which is otherwise obvious
THEOREM i0.
P
n
(dl,
For every partition
d
i
k)
of n,
a decomposition of
k).
i, 2,
into paths containing d. nodes (i
d
2,
The following corollary of Theorem 3 gives a recurrence which is useful for find-
ing the number of decompositions of a chain into paths with specified lengths.
COROLLARY 3.1.
Npn (n I
kI
k
n
PROOF.
2
2
n
kr)
r
kl
r
i--i
(n
Np
n-n
I
n
k2
2
ki-i
ni_ I
k-i
nil
i
From Theorem 3, we have
w P(n-l) + w P(n-2) +
P(n)
I
2
+
Wn.
k
i+l
ni+I
k
n
r
r)
528
E.J. FARRELL
k
NPn (nl
I
k
n2
k
k
2
r)
nr
is the coefficient of
I
Wnl Wn2
k
k
2
Wnr
only terms on the R.H.S. which contribute to this coefficient are w
Wn2P(n-n 2),
Npn
and
WnrP(n-nr )"
n
r
1
in P(n).
The
P(n-nl)
The result therefore follows.
Corollary 3.1 suggests an algorithm for finding the number
k
k
k
2
r
1
The following lemma is quite useful when applying Corol(n
n
I
2
nr
lary 3.1.
I.gNNA
Its proof is straightforward.
3.
(i)
Np (in22)
n
i.
Np (in33)
n
2.
n
(ii)
n
(iii)
(iv)
Npn (l,
n-l)
((n/k)
Np
2.
k)
i, where k divides n.
n
ILLUSTRATION.
NP6(I2, 22
NP5(l,
22
NP4(22
+
+
NP4 (12’ 2)
Np 3 (1,
2) +
Np 3 (i,
2) +
Np 2 (12
=1+2+2+1
6,
in agreement with Table I.
By a restricted decomposition of a graph G, we mean a decomposition in which particular paths must be included.
Clearly, the decompositions of restricted graphs will
be restricted.
The following corollaries can be easily deduced from the results given in Section
4 by putting w
i.
COROLLARY 5.2.
The number of restricted decompositions of the chain
which a particular terminal subchain
COROLLARY 6.2.
The number of restricted decompositions of
in which a particular subchain
included, is 2
parated by P
nl+n2-2
Pk
is always included, is 2
where n
Pn3
I
Pn+k
in
Pnl+n2+n3(nl,n2,n 3 > I)
(which does not include a terminal edge) is always
and n
2
are the numbers of nodes in the subchains se-
n3
and P
Let G be a graph consisting of 3 chains P ,P
n
n
I n2
3
> i) attached to a single node. The number of restricted decompositions of
3
COROLLARY 7.2.
(nl,n2,n
n-l.
G, in which the chain Pn
is always included, is 3.2
3
nl+n2-4
PATH DECOMPOSITIONS OF CHAINS AND CIRCUITS
COROLLARY 9.1.
529
Let G be the graph formed by attaching k chains to non-adjacent
nodes of valency 2 in P
The number of restricted decompositions of G, in which the
k
2
n
n-2k-l.
k chains are always included, is 3
By extracting the coefficient of w
Corollary 7.1 and the expression for
s
in the expression given for P(P
P(Pn
k*
*
nl,n 2
[r];w) in
;w) given in Theorem 9, we obtain the fol-
lowing results.
COROLLARY 7.3.
Let G be a graph consisting of 3 chains P
and P
n
2
3
The number of restricted decompositions of G, with
> I) attached to a node.
3
cardinality s(s > 1), in which the chain P
(nl,n2,n
where N
n
I
n
n
P
I
n
is always included, is
3
+ n 2.
COROLLARY 9.2.
Let G be a graph formed by attaching k chains to non-adjacent
nodes of valency 2 of a Ldin P
The number of restricted decompositions of G, with
n
cardinality s(s >k), in which the k chains are always included is
2
r=0
6.
k-r
k-r-1
PATH POLYNOMIALS OF CIRCUITS.
The notation C
n
will be used for a circuit with n nodes.
gives an expression for the path polynomial of C
THEOREM ii.
be P
rw
r=l
Apply the reduction process to C
G* will be C *[i].
n
P(n- r).
r
by deleting an edge
The graph
G"
will
Continue to apply the reduction process to the restricted
n
n
n
n
P(Cn;W)_
PROOF.
The following theorem
graphs formed, by always deleting an edge which is adjacent to an incorporated edge.
The graphs
edge of
Cn
G"
The graph formed by incorporating the final
will be restricted chains
cannot contribute to
P(Cn;W)_
since it contains an incorporated circuit.
By adding the contributions of the graphs
P(Cn;W)_
r=0
P(Pn_r* [r];w)_
G"
formed during the process, we get
(with
Pn*[0]
Pn
By using Theorem 5, we get
n-i
P(C ;w)
n
n-r
[
r=0
i=l
Wr+ i
P(n- r-i)
n
r w
r=l
r
P(n-r), as required.
1,2,3,4,5, and 6.
The following table gives values of C(n) (--P(C ;w)) for n
n
n
th
row is 2
i.
Observe that the sum of tile coefficients of the polynomial in the n
530
E. J. FARRELL
Tab le 2
Path Polynomials of Circuits
C (.n
n
i
w
2
w
3
w
I
2
+ 2w 2
3
+
I
I
3WlW2
+ 3w 3
w14 + 4w12w2 + 4WlW + 2w22 +
Wl + 13w2 + 5w12 + 5WlW2 + 5w2w3 + 5w5
+ 6w14w2 + 6w13w + 9w12w + 6w12w + 12WlW2W
+ 2w23 + 6w2w + 3w +
4
4w
3
5
5
w
5w
w
4
2
3
2
6
I
4
2
3
2
6w
4
3
+
6WlW 5
6
A generating function for P(C ;w) can be obtained by using a technique similar to
n
It is given in the following theorem.
that used in the proof of Theorem 4.
THEOREM 12.
P (C
where P(t)
n ;_,t)
2w2t2 + 3w3 t3+’’"
+
wit
w_
i-(w I
t
+
w2t
2
+
w3t
3
t
P’(t)
e(t)
+ ...)
P(Pn;W_,t).
[r] be a restricted circuit consisting of an incorporated path with r
n+r-i
edges and a path with n nodes. Let us apply the reduction process to C*
n+r- i [r] as we
did in the proof of Theorem ll for the restricted circuits. This will yield
n-i
Let C*
[r];w)
P (Cn+r-i
*
P
i=0
(Pn*-i [r +i];w)
By applying Theorem 5, we get the following result.
THEOREM 13.
e(Cn*+r_l[r];w)
n-i
n-i
i=O
j =i
Wr+i+j
P(n- i- j).
Consider now the case in which the incorporated chain P * is attached to C
We
r
n
will denote this graph by C**[r-i]. If we apply the reduction process to this graph,
n
by deleting an unincorporated edge which is adjacent to an incorporated edge, the
graph
G"
will be
P+r_l[r-i]
and G* will be P
*
q+r-I
[r].
Hence we obtain the following
result by using Theorem 5.
THEOREM 14.
n+r-i
P(Cn**[r- l];w)
7.
(Wr+ i
i=l
+ Wr+i_ I) P(n+r-
i- i).
SIMPLE PATH POLYNOMIALS OF CIRCUITS.
The simple path polynomial of C
n
can be easily deduced from Theorems 8 and ll,
PATH DECOMPOSITIONS OF CHAINS AND CIRCUITS
or by elementary combinatorial argument.
531
It is given in the following theorem.
THEOREM 15.
(1 + w)
P(C ;w)
n
C*n+r_l[r]
The restricted graph
porated path to a node.
1
z
n
1.
Cn_ 1
can be treated as
by "shrinking" the incor-
Thus we have
P(C*
[r];w)
n+r-1
COROLLARY 13.1.
n
P(en_l i
w)
z
n-I
i
Instead of deriving results for C**[r-1], we would solve the general problem in
n
n
which k incorporated chains are attached to non-adjacent nodes of C (k<_
]). Let
n
us denote this graph by e
n
We will consider 2 cases: (i) C
contains a pair of consecutive incorporated
n
chains which are separated by a path of length greater than 2, and (2) C
has no such
n
In case (1), we can apply the reduction process to C
pair of incorporated chains.
n
by deleting an edge which is adjacent to an incorporated chain and which belongs to a
"
ke
k*
k*
k*,
G"
separating path of length greated than 2.
will be P
n
(k+l),,
since the end-node to
which an incorporated chain is attached, can be treated as an ordinary node.
be
Pn_l (k-1)*.
G* will
Hence from Theorem 9, we get
nk
P(C *;w)
z
=z
n-2k-i
n-2k
z
nk
z
z
2
-1)
k
(z+l)
n-2k-i
n-2k-i
(z-I)
(z-l)
k- i
+
zn-2k
(z- i
)k
(z+l)
k-I
k
Pn_l (k-*
In case (2), G" will be
P(C *;w)
(z- i)
and G* will be
k
(z+l)
k-i
k-i
(z+l)
Pn-i
(k-2),
It follows that
+ zn-2k+2 (z-l) k-i (z+l) k-2
k-2
(z
3
+
z
2
i).
Hence we have the following theorem.
THEOREM 16.
by a path in C
n
If C
k*
n
contains a pair of consecutive incorporated chains, separated
of length greater than 2, (or if k
p(ck*;w)
n
z
n-2k
(z
i) then
l)k;
2
otherwise,
nk
P(C *;w)
8.
z
n-2k-I
(z-1
)k-i
(z+l) k-2
z
3
+
z
2
i).
PATH DECOMPOSITIONS OF CIRCUITS.
The covers of P
n
will also be covers of C
n
Therefore, it is expected that the
results similar to those given in Section 5 will also hold for circuits.
The following is an immediate corollary of Theorem 15.
It confirms our observa-
tion of the sum of the coefficients of the polynomials in Table 2.
COROLLARY ii.i.
Since the covers of P
following corollary.
N
n
C
=2n- 1
n
will also be covers of C
n
we have from Theorem i0, the
532
E.J. FARRELL
(dl, d2, d3,
For every partition
COROLLARY 10.1.
sition of C
dk)
of n,
3
a decompo-
1,2,...,k).
into paths containing d. nodes (i
i
n
The following corollary of Theorem ii is analogous to Corollary 3.1.
Its proof
is similar.
COROLLARY Ii.i.
k
k
1
2
N (n
n
2
C
I
k
n
n
r
r
r
n
i=l
i
Np
(nl
n-n.
kl
n2
k2
ki-1
ni-i
N
C
(1
n-3
ki-1
i
n
k
ki+ 1
n
i+l
r
r
It will be useful for practical
The following lemma is analogous to Lemma 3.
applications of Corollary 11.2.
n-2
LEMMA 4.
2)
(i) N C (i
n
n
3)
N
n
C
(l,n-l)
n
N
C
(n)
n
n
k)
(n/k)
n/k, where k divides n.
n
The following corollaries are immediate from Corollary 13.1.
(ii)
COROLLARY 13.2.
ticular subchain
Pk+2
N
C
The number of restricted decompositions of
is always included is 2
n-I
Cn+k
in which a par-
i.
COROLLARY 13.3.
The number of restricted decompositions with cardinality s, of
n-i
is always included is
)"
Cn+k,
s
From Theorem 16, we obtain the following corollaries.
in which a particular subchain
COROLLARY 16.1.
to it.
Pk+2
Let G be a graph consisting of a circuit with k chains attached
If G contains a pair of consecutive attached chains which are separated by a
path of length greater than 2, then the number of restricted decompositions of G in
k
2
which all k chains are included is 3
Otherwise, the number of such
k-2
2
3
restricted decompositions is ii
n-2k.
n-2k-l.
COROLLARY 16.2.
The number of restricted decompositions with cardinality s(s-k)
of the graph G of Corollary 16.1 is
k
r=0
rk
n-2k
(s_k_r) 2k-r
when G contains a pair of consecutive attached chains, separated by a path of length
greater than 2.
Otherwise,
k-2
i=O
where N
n-2k-I and m
the number of such decompositions is
N
.N+2
+ 3(.N+I) + [m+l )]
(k2) 2k-m-i[(m_2)
m
s- i-k.
REFERENCES
i.
FARRELL, E.J. On a general class of graph polynomials, J. Comb. Theory B 26
(1979) 111-122.
2.
HARARY, F.
Graph T.heo.ry, Addison-Wesley, Reading, Mass., 1969.
PATH DECO[POSITIONS OF CHAINS AND CIRCUITS
533
3.
GOODMAN, S.E. and HEDETNIEMI, S.T. On the Hamlltonlan completion problem, in:
R.A. Burl and F. Harary, eds., Graphs and Combinations (Sprlnger-Verlag,
Berlin, 1974), 262-272.
4.
BOESCH, F.T., CHEN, S. and McHUGH, J.A.M. On covering the points of a graph with
disjoint paths in: R.A. Burl and F. Harary, eds., Graphs and Combinations
(Sprlnger-Verlag, Berlin, 1974), 201-212.
SLATER, P.L.
65-74.
6.
READ, R.C.
Path coverings of the vertices of a tree, Discrete Math. 25
(1979),
Introduction to chromatic polynomials, J. Comb. Theory 4 (1968), 52-7L