Electronic Notes in Discrete Mathematics 22 (2005) 439–444
www.elsevier.com/locate/endm
Edge Addition Number of Cartesian Product of
Paths and Cycles
Yung-Ling Lai 1,2 Chang-Sin Tian 3 Ting-Chun Ko 4
Department of Computer Science and Information Engineering
National Chiayi University
Chiayi, Taiwan
Abstract
Graph bandwidth problem has been studies for over forty years. The relation between the bandwidth and number of edges in the graph is always interesting. The
edge addition number ad(G) of a graph is the minimum number of edges that added
into the graph G which cause the resulting graph’s bandwidth greater than the
bandwidth of G. This paper determines the edge addition number for the Cartesian
product of a path with a path, a path with a cycle and a cycle with a cycle.
Keywords: Bandwidth, Cartesian product, edge addition number.
1
Introduction
Edge addition is a local operation on a graph. Let G = (V, E) be a graph.
If u and v are nonadjacent vertices of G, then G + e where e = uv denotes
the graph obtained from G by adding edge e. Let X ⊆ E(G), then G + X
1
2
3
4
This research is partially supported by NSC-92-2115-M-415-001
Email: mailto:[email protected] [email protected]
Email: mailto:[email protected] [email protected]
Email: mailto:[email protected] [email protected]
1571-0653/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.endm.2005.06.062
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Y.-L. Lai et al. / Electronic Notes in Discrete Mathematics 22 (2005) 439–444
denote the graph obtained from G by adding all edges in the set X. Given
a graph G = (V, E), a proper labeling f of G is a bijection function f :
V → {1, 2, · · · , |V |}. The bandwidth of a proper labeling f of G is defined as
Bf (G) = max {|f (u) − f (v)| : uv ∈ E} and the bandwidth of G is the number
B(G) = min {Bf (G) : f is a proper labeling of G}. A proper labeling f is
called a bandwidth labeling of G if Bf (G) = B(G).
The bandwidth problem was first studied in 1950s, while they were trying to lay all the nonzero entries in the matrix within a narrow band about
the main diagonal (hence has the term ”bandwidth”), and graph bandwidth
problem independently from the matrix bandwidth in 1960s where they were
trying to minimize the maximum absolute error in the picture code which was
represented by a hypercube [9]. In 1976, Papadimitriou [14] proved that the
decision problem of graph bandwidth is NP-Complete, in 1978, Garey et. al.
[8] proved that the problem stays in NP-Complete even with restriction of
graph as tree with maximum degree 3. A large number of relevant problems
in different domains can be formulated as graph bandwidth problem, such as
solving linear equations, optimization of networks for parallel computer architectures, VLSI layout, constraint satisfaction problem, computational biology
and scheduling, etc. Because of the importance of graph bandwidth, there is
a strong interesting in this area since mid-sixties. There are several survey
papers in this area, see for instance Chinn et. al. [2] at 1982, Lai and Williams
[13] at 1999, and Diaz et. al. [6] at 2002.
2
Related Works
By adding an edge to graph G, it is trivial that B(G + e) ≥ B(G). We know
that when G is a cycle or a complete bipartite graph, B(G + e) = B(G),
and when G is a path, B(Pn + e) = B(Pn ) + 1, no matter which edge is
selected. Chvátalová [4] showed that B(G + e) − B(G) may be greater than
1. Chvátalová and Opatrný [5] provided the upper bound for bandwidth after
an edge addition as B(G + e) ≤ 2B(G) which can be further improved when
e = uv with d(u, v) ≤ 3 as B(G + uv) ≤ (d(u, v) + 2)/3 B(G). In 1995,
Wang [15] determined the upper bound of B(G + e) in terms of B(G) and |V |.
In 1989 and 1992, Dutton and Brigham [7] and Alavi [1] established some
bounds for the size of graph with given bandwidth. Lai [11] defined the edge
addition number of G to be the number
ad(G) = min{|X| : X ⊆ E(G) such that B(G + X) > B(G)}
and obtained ad(Cn ) for n ≥ 4 and ad(Km,n ) for 1 ≤ m ≤ n.
Y.-L. Lai et al. / Electronic Notes in Discrete Mathematics 22 (2005) 439–444
441
The Cartesian product of two graphs G and H, denoted G × H, is the
graph with vertex set V (G) × V (H) and (u1 , v1 ) is adjacent to (u2 , v2 ) if
either u1 is adjacent to u2 in G and v1 = v2 or u1 = u2 and v1 is adjacent
to v2 in H. The Cartesion product of paths and cycles are well studied.
In [3], they gave B(Pm × Pn ) = m for m ≤ n; [10] and [12] established
B(Cm × Cn ) = 2m for 3 ≤ m < n and B(Cm × Cm ) = 2m − 1 for m ≥ 3; and
B(Pm × Cn ) = min{2m, n} was provided by [3], [4], [12], [16]. In this paper,
we determine ad(G) where G is Pm × Pn , Cm × Cn , or Pm × Cn for m ≤ n.
3
Main Results
Theorem 3.1 Let G = Pm × Pn where 2 ≤ m ≤ n. Then
⎧
⎨ 1, if 2 ≤ m < n or 6 ≤ m ≤ n;
ad(G) =
⎩ 2, if 2 ≤ m = n ≤ 5.
Proof. For m < n, we only need to add e = v1,1 v1,n , then B(G + e) > m =
B(G). For 6 ≤ m = n, let e = v1,1 vn,3 then B(G + e) > m = B(G). When
m = n = 2, G = C4 which implies ad(G) = 2(by [11]). For 3 ≤ m = n ≤ 5, we
have to add X = {v1,1 v1,m , v1,1 vm,1 } then B(G + X) > m = B(G). To see that
ad(G) > 1 when 3 ≤ m = n ≤ 5, if we add an edge in any row (or column),
a row by row (column by column) labeling will be a bandwidth labeling of
the new graph as well as of G. If e = vi,j vk,l where i = k and j = l, then a
diagonal labeling will give B(G + e) = B(G). Therefore, ad(G) > 1.
2
For torus, next two theorems discuss different cases of Cm ×Cn . In theorem
3.2 we discuss the Cartesian product of two different cycles and theorem 3.3
we discuss the case of two equal cycles.
Theorem 3.2 Let G = Cm × Cn where 3 ≤ m < n. Then
⎧
⎨ 6, if n = 4;
ad(G) =
⎩ 2, otherwise.
Proof. G = C3 × C4 is a special case where we have to add 2 edges in
the 4-cycle of each row, which results the graph C3 × K4 . Since we know
that B(C3 × K4 ) = 7 > 6 = B(C3 × C4 ) and there are 3 rows, therefore
ad(C3 × C4 ) = 6.
For n = 4, it is easy to see that ad(G) ≥ 2 since adding any edge
e ∈ E(G) to G we can always relabel the graph to make the edge e does
not effect to the bandwidth of G + e. To see that ad(G) ≤ 2, let X =
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Y.-L. Lai et al. / Electronic Notes in Discrete Mathematics 22 (2005) 439–444
j
j
j
j
j
j
Fig. 1. Graph H
(v1,1 v1,(n+1)/2 ), (v1,(n+1)/4 v1,(3n+1)/4 ) . Then we have B(G + X) > 2m =
B(G), which implies the result.
2
Theorem 3.3 Let G = Cm × Cm . Then
⎧
⎨ 6, if m = 3;
ad(G) =
⎩ 4, if m ≥ 4.
Proof. G = C3 × C3 is a special case since there is no edge we can add into
the original cycle. In this case, since B(G) = 5, B(G + X) > B(G) if and only
if G + X does not contain H (see Fig. 1) as a subgraph. Since all vertices in
G are similar vertices, adding any 5 or less edges we are able to re-labeling the
graph to make B(G + X) = 5, so ad(G) ≥ 6. To see that ad(G) ≤ 6, consider
X = {(v1,1 v2,2 ), (v2,1 v1,2 ), (v2,1 , v3,3 ), (v3,1 v2,3 ), (v1,2 v3,3 ), (v3,2 v1,3 )}. Then we
have B(G + X) ≥ 6 > 5 = B(G). Hence we have ad(C3 × C3 ) = 6.
For m ≥ 4, we have to add 2 edges in one of the row cycle and 2 edges in
one of the column cycle to make the resulting graph has greater bandwidth
than B(G). Let t = (m + 1)/2, consider
X = {(v1,1 vt,t ), (v1,t vt,1 ), (v1,2 vt,t+1 ), (vt,2 v1,t+1 )} ,
then |X| = 4 and we have B(G + X) > 2m − 1 = B(G), which implies the
result.
2
Theorem 3.4 Let G = Pm × Cn for 3 ≤ m ≤ n. Then
⎧
⎪
⎪
1, if 2m > n;
⎪
⎨
ad(G) = 2, if 2m < n;
⎪
⎪
⎪
⎩ 3, if 2m = n.
Proof. For the case 2m > n, Since B(G) = n, consider adding the edge
e = v1,1 vm,1 , then we have B(G + e) > n = B(G). Now consider the case
2m < n, since B(G) = 2m, similar
to the case of theorem 3.2, consider adding
the set X of edges where X = (v1,1 v1,(n+1)/2 ), (v1,(n+1)/4 v1,(3n+1)/4 ) , then
Y.-L. Lai et al. / Electronic Notes in Discrete Mathematics 22 (2005) 439–444
443
we have B(G +X) > B(G). When 2m = n, similar to theorem
3.3, after adding X = (v1,1 v1,(n+1)/2 ), (v1,(n+1)/4 v1,(3n+1)/4 ), (v1,1 vm,1 ) to G, we
have B(G + X) > B(G).
2
4
Conclusion
The bandwidth problem of Cartesian product of paths and cycles has shown
it’s interesting in the eyes of researchers. People are also interested in the
graph bandwidth problem after adding edges to the graphs. This paper provides the edge addition number of Cartesian product of paths and cycles,
which shows the minimum number of edges one has to add into such graphs
in order to increase the bandwidth in the resulting graph.
References
[1] Alavi, Y., J. Liu, and J. McCanna, On the minimum size of graphs with a given
bandwidth, Bulletin of the ICA 6 (1992), 22-32.
[2] Chinn, P.Z., J. Chvátalová, A.K. Dewdney, and N.E. Gibbs, The bandwidth
problem for graphs and matrices—A survey, J. Graph Theory 6 (1982), 223254.
[3] Chvátalová, J., Optimal labeling of a product of two graphs, Discrete math 11
(1975), 249-253.
[4] Chvátalová, J., On the bandwidth problem for graphs, Ph.D. thesis, University
of Waterloo, Canada, 1980.
[5] Chvátalová, J., J. Opatrny, The bandwidth problem and operations on graphs,
Discrete Math. 61 (1986), 141-150.
[6] Diaz, J., J. Petit and M. Serna, A survey of graph layout problems, ACM
Computing Surveys 34(3) (2002), 313-356.
[7] Dutton, R. D., and R. C. Brigham, On the size of graphs of a given bandwidth,
Discrete Mathematics 76 (1989), 191-195.
[8] Garey, M. R., R. L. Graham, D. S. Johnson and D. E. Knuth, Complexity
results for bandwidth minimization, SIAM Journal on Applied Mathematics,
(1978), 477-495.
[9] Harper, L. H, Optimal assignment of numbers to vertices, J. SIAM 12 (1964),
131-135.
444
Y.-L. Lai et al. / Electronic Notes in Discrete Mathematics 22 (2005) 439–444
[10] Hendrich U., and M. Stiebitz, On the bandwidth of graph products, J Info Proc
Cyber 28 (1992), 113-125.
[11] Lai, Y. L., C. S. Tain, and T. C. Ko, Adding minimum number of edges to
increase the bandwidth of graphs, The proceeding of the 2nd International
Conference on Research and Education in Mathematics (ICREM 2), Serdang,
MALAYSIA, (2005), 504-509.
[12] Lai Y. L., and K. Williams, On bandwidth for the tensor product of paths and
cycles, Discrete Applied Mathematics 73 (1997), 133-141.
[13] Lai Y. L., and K. Williams, A survey of solved problems and applications on
bandwidth, edgesum, and profile of graphs, Journal of Graph Theory 31(2)
(1999), 75-94.
[14] Papadimitriou, C. H., The NP-completeness of the bandwidth minimization
problem, Computing, 16 (1976) 263-270.
[15] Wang, J.-F., D.B. West, B. Yao, Maximum bandwidth under edge addition, J.
Graph Theory 20 (1995), 87-90.
[16] Zabih, R., Some applications of graph bandwidth to constraint satisfaction
problems, AAAI-90: Proc Eighth Nat Conf Art Intell MIT, Cambridge, MA
(1990), 46-51.