In this section, we use almost the same idea of the previous sections to establish the bandwidths of G(N, D).
2.4.1 Bandwidths on G(N, D), G(N, D)H, G(N, D)[H], and G(N, D) ∧ H
Theorem 2.4.1 B(G(N, D)) = λ.
Proof. First, consider the identity numbering id from V (G(N, D)) to N. Then we have B(G(N, D)) ≤ Bid(G(N, D)) = λ. Next, we show that λ is the upper bound. Let f be a bandwidth numbering on G(N, D), and let t = max{f (i) : 1 ≤ i ≤ λ}. Since t is finite, for i ∈ [λ], there is i + kiλ = min{i + kλ : f (i + kλ) ≥ t + 1, k ∈ N}. It means that i + kiλ ∈ ∂Stf for i ∈ [λ]. Then by Proposition 2.1.3, we get B(G(N, D)) ≥ λ, as desired.
Theorem 2.4.2 B(G(N, D)H) = mλ.
Proof. Consider the numbering g from V (G(N, D)H) to N defined by g(vi,j) = (i − 1)m + j. Obviously, B(G(N, D)H) ≤ Bg(G(N, D)H) = mλ. Next, we need to show that B(G(N, D)H) ≥ mλ. Let f be a bandwidth numbering on G(N, D)H, and let t = max{f (vi,j) : 1 ≤ i ≤ λ, 1 ≤ j ≤ m}. Since t is finite, for each (i, j) ∈ [λ] × [m], there is i + ki,jλ = min{i + kλ : f (vi+kλ,j) ≥ t + 1, k ∈ N}. It means that f (vi+(ki,j−1)λ,j) ≤ t, and so vi+ki,jλ,j ∈ ∂Stf for each (i, j) ∈ [λ] × [m]. For each j, let i + ki,jλ = i0+ ki0,jλ, where i, i0 ∈ [λ]. As 0 ≤ i − i0 = (ki0,j − ki,j)λ < λ, it forces i = i0 and ki,j = ki0,j. And hence ∂Stf ≥ mλ. By Proposition 2.1.3, we have B(G(N, D)H) ≥ ∂Stf ≥ mλ.
Theorem 2.4.3 B(G(N, D)[H]) = mλ + m − 1.
Proof. Consider the numbering g from V (G(N, D)[H]) to N defined by g(vi,j) = (i − 1)m + j. Then B(G(N, D)[H]) ≤ Bg(G(N, D)[H]) = m(λ + 1) − 1. Next, we have to show that B(G(N, D)[H]) ≥ mλ + m − 1. Let f be a bandwidth numbering on G(N, D)[H], and let
t = max{f (vi,j) : 1 ≤ i ≤ λ, 1 ≤ j ≤ m} ∪ {f (vλ+1,1)}, ti = min{f (vi,j) : 1 ≤ j ≤ m} for i ∈ N.
Define r(vi,j) = i and let ϑ = max{r(f−1(x)) : 1 ≤ x ≤ t}. Since t is finite, min{ti : ti ≥ t, i ≥ ϑ} exists, say t0 or f (vi0,j0). Also, because t0 is finite, for each (i, j) ∈ [λ] × [m], there is i + ki,jλ = min{i + kλ : f (vi+kλ,j) ≥ t0 + 1, k ∈ N}. Evidently, i + ki,jλ ≤ i0 + λ − 1 for each (i, j) ∈ [λ] × [m] but j = j0. Let L = {vi+ki,jλ,j : 1 ≤ i ≤ λ, 1 ≤ j ≤ m}, by the same argument as in Theorem 2.4.2, it is known that L has mλ vertices in ∂Stf0. Besides, for j 6= j0, vi0+λ,j ∈ ∂Stf0 \ L by the definition of t0. So L ∪ {vi0+λ,j : j 6= j0} ⊆ ∂Stf0, and hence ∂Stf0
≥ mλ + (m − 1).
Theorem 2.4.4 B(G(N, D) ∧ H) = (m + 1)λ.
Proof. Let G0 = G([n], D) ∧ H. Since D(G0) ≤n−1
λ
+ ε(λ − 1) + 2, from the Proposition 2.1.4, we have
B(G(N, D) ∧ H) ≥ n(m + 1) − 1
n−1
λ
+ ε(λ − 1) + 2
≥ n(m + 1) − 1
n−1
λ + ε(λ − 1) + 2
= λn(m + 1) − λ n − 1 + ελ(λ − 1) + 2λ
= (m + 1)λ − λn 1 + ελ(λ−1)+2λ−1
n
.
Take limitation on such n that G([n], D) ∧ H is a connected subgraph of G(N, D) ∧ H, and by Proposition 2.1.1, there is no doubt that B(G(N, D) ∧ H) ≥ (m + 1)λ. Next, to show (m + 1)λ is an upper bound of B(G(N, D) ∧ H), we consider a numbering g of G(N, D) ∧ H by
g(i) = (i − 1)m + i, for i ∈ N;
(i − 1)m + i + 1 ≤ g(v) ≤ (i − 1)m + i + m, for v is in the copy of H corresponding to i.
Now if two vertices x and y are in the same component {i} ∨ Hi, then we have |g(x) − g(y)| ≤ m. The only other vertices adjacent in G(N, D) ∧ H are those which are adjacent in G(N, D). Assume i is adjacent to j in G(N, D). Then |g(i) − g(j)| = |(i − j)(m + 1)| ≤ (m + 1)λ. These give B(G(N, D) ∧ H) ≤ Bg(G(N, D) ∧ H) = (m + 1)λ. (In fact, it is easy to prove that B(G ∧ H) ≤ B(G)|V (H)| for arbitrary graphs G and H.)
2.4.2 Bandwidths on G(N, D) × H and G(N, D) H
Define two parameters as
Bp(H; k) = min
|A|=k
∪v∈AN (v)
− k : A ⊆ V (H)
, Bp(H) = max
k Bp(H; k).
We may use them to express a lower bound of B(G(N, D) × H) as follows.
Proposition 2.4.5 Let H be a Hamiltonian graph or have a perfect matching, and let f be a bandwidth numbering on G(N, D) × H. Then there is a t ∈ N such that ∂Stf ≥ mλ + Bp(H), and therefore B(G(N, D) × H) ≥ mλ + Bp(H).
Proof.
Either of them forces inconsistencies or i = k, j = l.
And let
µ = max{f (vi,j) : 1 ≤ i ≤ λ, 1 ≤ j ≤ m}, c(vi,j) = j,
Aρ,h =
yj ∈ V (H) : j ∈ ∪
i≥hc(f−1([ρ]) ∩ Ri)
for ρ ≥ µ and h ≥ λ + 1, t = min {ρ : |Aρ,h| = `} ,
ht = min {h : |At,h| = `} , W =
vi(j),j : j ∈ ∪
i≥ht
c(f−1([t]) ∩ Ri), i(j) = max r(f−1([t]) ∩ Cj)
.
For 1 ≤ i ≤ λ, 1 ≤ j ≤ m, let
i + 2ai,jλ = min{i + 2θλ : f (vi+2θλ,j) ≥ t + 1, θ ∈ N },
i + (2bi,j − 1)λ = min{i + (2θ − 1)λ : f (vi+(2θ−1)λ,j+(−1)j+1) ≥ t + 1, θ ∈ N }, f
vi,j =
vi+2ai,jλ,j, for 2ai,j < 2bi,j− 1;
vi+(2bi,j−1)λ,j+(−1)j+1, for 2ai,j > 2bi,j− 1.
And let
T = { fvi,j : 1 ≤ i ≤ λ, 1 ≤ j ≤ m}, T0 =
vi(j)+λ,n(j) : vi(j),j ∈ W, yn(j)∈ NH(yj) .
With the similar argument of Case 1, we also get
∂Stf ≥ mλ + maxk
|A|=kmin
∪v∈AN (v)
− k : A ⊆ V (H)
= mλ + Bp(H).
In more general, we have the following consequences by a careful application of Propo-sition 2.4.5.
Theorem 2.4.6 If a graph H has a spanning subgraph which consists of a disjoint union of cycles or a matching, then B(G(N, D) × H) ≥ mλ + Bp(H).
We next give a weaker lower bound of B(G(N, D) × H) which is easy to obtain from Theorem 2.4.6.
Corollary 2.4.7 If a graph H has a spanning subgraph which consists of a disjoint union of cycles or a matching, then B(G(N, D) × H) ≥ mλ + δ(H) − 1.
Proof. Taking |A| = 1 in Theorem 2.4.6, we then have this corollary.
Lemma 2.4.8 B(G(N, D) × H) ≤ mλ + B(H) for any finite graph H.
Proof. Let f be a bandwidth numbering of H. Consider the numbering g : V (G(N, D) × H) → N defined by g(vi,j) = (i − 1)m + f (yj). It is clear that B(G(N, D) × H) ≤ Bg(G(N, D) × H) = mλ + B(H).
We give exact values of bandwidth for some G(N, D) × H0s below.
Example 2.4.9
(1)B(G(N, D) × Pm) = mλ + 1 for m ∈ 2N \ {2}.
(2)B(G(N, D) × Cm) = mλ + 2 for m ≥ 4.
Proof. It is trivial to get their upper bounds from Lemma 2.4.8. We know their lower bounds by taking |A| = m − 1 for (1) and |A| = m − 2 for (2) in Theorem 2.4.6. Thus the results hold.
Figure 2.9 shows a bandwidth numbering of G(N, D) × C5 with max D = 3 in which the edges are not drawn completely.
t
For a graph H obtained from join, we give another substitutional bounds.
Theorem 2.4.10 Let Hr be a graph of order mr for r ∈ [t] and H = ∨
1≤r≤tHr of order
m = X
1≤r≤t
mr. If H has a spanning subgraph which consists of a disjoint union of cycles or a matching, then
Proof. By Theorem 2.4.6, we know
B(G(N, D) × H) ≥ mλ + max
If not, then
Next, we need to show that mλ+ max
1≤r≤t(m−mr+B(Hr)) is an upper bound of B(G(N, D)×
Proof. Since H can be spanned by a disjoint union of some cycles and a matching, the corollary follows from Theorem 2.4.10.
Example 2.4.12 for m ≥ 4, we have those consequences from the above corollary.
Corollary 2.4.13 Let H0 be a graph of order m0 ≤ m + δ(H0). If H = Km ∨ H0 can be spanned by a disjoint union of some cycles and a matching, then B(G(N, D) × H) = (m + m0)λ + m0− 1.
Proof. By Theorem 2.4.10 and m0 ≤ m + δ(H0), we know
B(G(N, D) × H) ≥ (m + m0)λ + max
k
1≤r≤tmin
m − mr+ Bp(Hr; k)
≥ (m + m0)λ + minn
m0+ Bp(Km; 1), m + Bp(H0; 1)o
= (m + m0)λ + min {m0+ (−1), m + (δ(H0) − 1)}
= (m + m0)λ + m0− 1.
Next, we need to show that (m + m0)λ + m0− 1 is an upper bound of B(G(N, D) × H).
Let V (Km) = {yj : 1 ≤ j ≤ m} and V (H0) = {yj : m + 1 ≤ j ≤ m + m0}. Consider a numbering g from V (G(N, D) × H) to N defined by
g(vi,j) = (i − 1)(m + m0) + j for (2k − 2)λ + 1 ≤ i ≤ (2k − 1)λ,
and for (2k − 1)λ + 1 ≤ i ≤ 2kλ
g(vi,j) =
(i − 1)(m + m0) + j + m0, for 1 ≤ j ≤ m;
(i − 1)(m + m0) + j − m, for m + 1 ≤ j ≤ m + m0,
where k ∈ N. It is not hard to check that B(G(N, D) × H) ≤ Bg(G(N, D) × H) = (m + m0)λ + m0− 1.
Example 2.4.14 B(G(N, D) × Kt(m)) = tmλ + (t − 1)m − 1 for t ≥ 2.
Proof. We obtain the result immediately by Corollary 2.4.13. Notice that Km(1) means Km. This implies B(G(N, D) × Km) = mλ + m − 2.
Figure 2.10 shows a bandwidth of G(N, D) × K3,3 with max D = 2 in which the edges are not drawn completely.
t
In the following, we imitate the process of argument on B(G(N, D)×H) to give similar results of B(G(N, D) H). Also, first of all, we define two parameters as
Bs(H; k) = min
Proof. The upper bound can be easily derived by the same argument as in the proof of Lemma 2.4.8. As to the lower bound, we discuss it in detail below. Suppose f is a bandwidth numbering on G(N, D) H. Let A0 = {yjs : 1 ≤ s ≤ `} ⊆ V (H) such that
And let
We next also give a weaker lower bound of B(G(N, D) H) which is easy to obtain from Theorem 2.4.15.
Corollary 2.4.16 B(G(N, D) H) ≥ mλ + δ(H).
Proof. Taking |A| = 1 in Theorem 2.4.15, we then have the corollary.
We also offer exact values of bandwidth for some G(N, D) H’s in the underside.
Example 2.4.17
(1)B(G(N, D) Pm) = mλ + 1.
(2)B(G(N, D) Cm) = mλ + 2.
(3)B(G(N, D) Km) = mλ + m − 1.
Proof. The results follow from Theorem 2.4.15.
For a graph H obtained from join, we still give another substitutional bounds.
Theorem 2.4.18 If Hr is a graph of order mr for r ∈ [t] and H = ∨
Proof. By Theorem 2.4.15, we know
B(G(N, D) H) ≥ mλ + max
Next, we need to show that mλ+ max
1≤r≤t(m−mr+B(Hr)) is an upper bound of B(G(N, D)
f (yj) = fr(yj) for X
1≤s≤r−1
ms+ 1 ≤ j ≤ X
1≤s≤r
ms. Suppose i = λai+ bi for each i ∈ N, where bi ∈ [λ]. Consider a numbering g from V (G(N, D) H) to N defined by
(i − 1)m + 1 ≤ g(vi,j) ≤ im, and g(vi,j) ≡ f (yj) − X
1≤s≤ai
ms (mod m).
It is not hard to check that B(G(N, D) H) ≤ Bg(G(N, D) H) = mλ + max
1≤r≤t(m − mr+ B(Hr)).
Corollary 2.4.19 If Hr is a graph of order m for all r ∈ [t] and H = ∨
1≤r≤tHr, then maxk
1≤r≤tmin Bs(Hr; k)
≤ B(G(N, D) H) − [tmλ + (t − 1)m] ≤ max
1≤r≤tB(Hr).
Proof. We may get this result directly from Theorem 2.4.18.
Example 2.4.20 (1)B(G(N, D) ( ∨
1≤r≤tPm)) = tmλ + (t − 1)m + 1 for m ≥ 3.
(2)B(G(N, D) ( ∨
1≤r≤tCm)) = tmλ + (t − 1)m + 2 for m ≥ 4.
Proof. Since max
k Bs(Pm; k) = 1 = B(Pm) for m ≥ 3 and max
k Bs(Cm; k) = 2 = B(Cm) for m ≥ 4, we have these consequences from the above corollary.
Corollary 2.4.21 If H0 is a graph of order m0 ≤ m + δ(H0) and H = Km ∨ H0, then B(G(N, D) H) = (m + m0)λ + m0.
Proof. By almost the same argument as in Corollary 2.4.13, we have the result.
Example 2.4.22 B(G(N, D) Kt(m)) = tmλ + (t − 1)m for t ≥ 2.
Proof. We obtain it immediately by Corollary 2.4.21. Notice that Km(1) means Km. This implies B(G(N; D) Km) = mλ + m − 1.