Graphs and Combinatorics

Graphs and Combinatorics (2001) 17:775±784

Graphs and Combinatorics Ó Springer-Verlag 2001

The Hamiltonian Index of a Graph* Liming Xiong Department of Mathematics, Jiangxi Normal University, Nanchang 330027, P.R. China e-mail: [email protected]

Abstract. It is proved that the hamiltonian index of a connected graph other than a path is less than its diameter which improves the results of P. A. Catlin etc. [J. Graph Theory 14 (1990) 347±364] and M. L. Sarazin [Discrete Math. 134(1994)85±91]. Nordhaus-Gaddum's inequalities for the hamiltonian index of a graph are also established. Key words. Hamiltonian index, Diameter, Nordhaus-Gaddum's inequality

1. Introduction The graphs considered in this paper are ®nite undirected graphs and allowed to have multiple edges but no loops. The multigraph of order 2 with two edges will be called a 2-cycle and is regarded as hamiltonian. The line graph L…G† of a graph G has E…G† as its vertex set and two vertices are adjacent in L…G† if and only if they are adjacent as edges in G. The n-th iterated line graph Ln …G† is de®ned recursively by L0 …G† ˆ G and Lk‡1 …G† ˆ L…Lk …G††…k 2 N ˆ f0; 1; 2; . . .g†, where L1 …G† ˆ L…G† and Lk …G† is assumed to be nonempty. The hamiltonian index of a graph G, denoted by ham…G†, is de®ned to be ham…G† ˆ minfn : Ln …G† is hamiltoniang: Since the hamiltonian index does not exist for paths and is obviously 0 for cycles, we will exclude them from consideration in the rest of the paper. Let H be a subgraph of a graph G ˆ …V ; E†. We use the symbols e…H †, D…H † and d…H † to denote the number of edges, the maximum degree and the minimum degree of H respectively. We use NH …u† denotes the set of neighbors of a vertex u in H . The degree dH …v† of a vertex v in H is the number of edges of H incident with v. De®ne Vi …G† ˆ fv 2 V …G† : dG …v† ˆ ig and W …G† ˆ V …G†nV2 …G†. A branch in G is a nontrivial path whose ends are in W …G† and whose internal vertices, if any, have degree 2 (and thus are not in W …G†). If a branch has length one, then it has no internal vertex. We denote by B…G† the set of branches of G and by B1 …G† the * This research is supported by Natural Science Fund of the Province of Jiangxi

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subset of B…G† in which every branch has an end in V1 …G†. For any subgraph H of G, we denote by BG …H † the set of branches of G whose edges are all in H . Let G H denote the induced subgraph G‰V …G†nV …H †Š of G, i.e., the subgraph obtained from G by deleting the vertices in H together with their incident edges. For a nonempty subset F of E…G†, let G‰F Š denote the edge-induced subgraph, i.e., the subgraph of G whose vertex set is the set of ends of edges in F and whose edge set is F . If F1 and F2 are two subsets of E…G†, then H ‡ F1 F2 denotes the subgraph ofSG obtained from G‰…E…H † [ F1 †nF2 Š by adding to its vertex set any D…G† vertices of iˆ3 Vi …G† which were not already in its vertex set, and that any vertices so added are to be isolated vertices in H ‡ F1 F2 . A subgraph C of G is called a circuit if C is a connected subgraph in which every vertex has even degree. A graph G is supereulerian if G has a circuit which contains all vertices of G. The distance dH …G1 ; G2 † between two subgraphs G1 and G2 of H is de®ned to be minfdH …u1 ; u2 † : u1 2 V …G1 † and u2 2 V …G2 †g where dH …u1 ; u2 † denotes the length of a shortest path from u1 to u2 in H . If G1 is an subgraph of H induced by exactly one edge e, then we replace dH …G1 ; G2 † by dH …e; G2 †. The diameter of a subgraph H , denoted by dia…H †, is de®ned to be maxfdH …u; v† : u; v 2 V …H †g. Harary and Nash-Williams characterized the hamiltonian line graph as follows. Theorem 1.1 [4]. Let G be a connected graph. Then the line graph L…G† is hamiltonian if and only if G has a circuit C such that E…G† ˆ fe 2 E…G† : dG …e; C† ˆ 0g. Recently, L. Xiong and Z. Liu characterized those graphs G for which Ln …G†…n  2† is hamiltonian. Theorem 1.2 [8]. Let G be a connected graph and n  2. Then the n-th iterated graph Ln …G† is hamiltonian if and only if En …G† 6ˆ /, where En …G† denotes the set of those subgraphs H of a graph G which satisfy the following conditions: (i) (ii) (iii) (iv) (v)

any vertex H has even degree in H, i.e., dH …x†  0…mod2† for any x 2 V …H †; Sof D…G† V0 …H †  iˆ3 Vi …G†  V …H †; dG …H1 ; H H1 †  n 1 for any subgraph H1 of H ; jE…b†j  n ‡ 1 for any branch b 2 B…G†nBG …H †; jE…b†j  n for any branch b 2 B1 …G†.

In Section 2, we will show that the hamiltonian index of a connected graph other than a path is less than its diameter which improves the results of P. A. Catlin etc. [1] and M. L. Sarazin [6]. In Section 3, we will consider the hamiltonian index of the complement of a graph. 2. A Sharp Upper Bound In this section, our main purpose is to improve the following result. Theorem 2.1 [1]. Let G be a connected graph that is neither a path nor a 2-cycle. Then ham…G†  dia…G†:

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Using the reduction method, M. L. Sarazin [6] recently gave a sharp upper bound for the hamiltonian index of a connected simple graph other than a path. Theorem 2.2 [6]. Let G be a connected simple graph of order n other than a path. Then ham…G†  n

D…G†:

One can easily see that dia…G† 1  n D…G† [9]. Using Theorem 1.2, L. Xiong et al. [8] gave a simple proof of Theorem 2.2 and here we improve it to Theorem 2.4. The related result was obtained by Veldman [7]. Theorem 2.3 [7]. Let G be a connected graph with at least three edges and such that dia…G†  2. Then L…G† is hamiltonian, i.e., ham…G†  1. Theorem 2.4. Let G be a connected graph other than a path. Then ham…G†  dia…G†

1

and the upper bound is sharp. Proof of Theorem 2.4. We only need to consider the case that dia…G†  3 since G is a complete graph if dia…G† ˆ 1 and ham…G†  S 1 if dia…G† ˆ 2 by Theorem 2.3. D…G† De®ne H0 as the subgraph of G having V …H0 † ˆ iˆ3 Vi …G† and E…H0 † ˆ /. And then de®ne H as subgraph of G obtained from H0 by adding to H0 as many vertices of V2 …G† as possible (along with the least number of appropriate edges) so that H satis®es both (i) and (ii). Next we will prove that H 2 Edia…G† 1 …G†. We only need to prove the following three claims. Claim 1. dG …H1 ; H

H1 †  dia…G†

2 for any subgraph H1  H .

Proof of Claim 1. Let p ˆ u1 u2


us be a shortest path from H1 to H H1 with u1 2 V …H1 † and us 2 V …H H1 †. Obviously fu2 ; u3 ; . . . ; us 1 g \ V …H † ˆ /. By H satisfying (ii), fu2 ; u3 ; . . . ; us 1 g  V2 …G†

…1†

Suppose dG …H1 ; H H1 †  dia…G† 1, i.e., e…p†  dia…G† 1  2. Since H satis®es both (i) and (ii) and s ˆ e…p† ‡ 1  3; fu1 ; us g \ V2 …G† ˆ / and p 2 B…G†nB1 …G†. Hence, we have fu1 ; us g 

D…G† [ iˆ3

Vi …G†

…2†

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By (1) and (2), we can select two edges e1 ˆ u1 x and e2 ˆ us y of G such that x 6ˆ y and fx; yg \ V …p† ˆ /. Let p0 be a shortest path from x to y in G. Obviously e…p† 2  e…p0 †  e…p† ‡ 2. If e…p0 † ˆ e…p† ‡ 2, then e…p0 † ˆ dG …H1 ; H H1 † ‡ 2  dia…G† ‡ 1, a contradiction. If e…p0 † ˆ e…p† 2, then u1 p0 us is a shortest path from H1 to H H1 . By H satisfying (ii), V …p0 †  V2 …G†. Therefore, by (1), H 0 ˆ H ‡ E…p† [ E…u1 p0 us † is a subgraph of G which satis®es both (i) and (ii) but it contains more vertices in V2 …G† than H does, a contradiction. So we can assume e…p† 1  e…p0 †  e…p† ‡ 1. Hence V …p† \ V …p0 † ˆ /; jV …p0 † \ V2 …G†j  e…p†

1 and jV …p0 † \ V …H †j  3

…3†

We will obtain contradictions by considering the following cases. Case 1. e…p†  3. By (1), (2) and (3), H 0 ˆ H ‡ …E…p† [ E…u1 p0 us ††

…E…u1 p0 us † \ E…H ††

is a subgraph of G which satis®es both (i) and (ii) but it contains more vertices in V2 …G† than H does, a contradiction. Case 2. e…p† ˆ 2. By (1), (2) and (3), jV …p0 † \ V2 …G† \ V …H †j  2: We will distinguish the following two subcases. Case 2.1. jV …p0 † \ V2 …G† \ V …H †j  1: By (1), (2) and (3), H 0 ˆ H ‡ …E…p† [ E…u1 p0 us ††

…E…u1 p0 us † \ E…H ††

is a subgraph of G which satis®es both (i) and (ii) but it contains more vertices than H does, a contradiction. Case 2.2. jV …p0 † \ V2 …G† \ V …H †j ˆ 2: By (1), (2) and (3),

jV …p † \ V2 …G†j ˆ 3 and V …p0 † \ 0

D…G† [ iˆ3

! Vi …G† ˆ 1

…4†

0 Without SD…G† loss of generality, we let p ˆ xx1 x2 y and fx; x1 g  V2 …G† \ V …H †; x2 2 iˆ3 Vi …G†  V …H †; y 2 V2 …G†nV …H †. Hence by (2), there exists a vertex y 0 2 V …G† such that u3 y 0 2 E…G† and y 0 2 =fy; u2 g.

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Hence since dG …x; y 0 †  dia…G†  s ˆ 3, either x2 y 0 2 E…G† or dG …u1 ; y 0 †  2. If x2 y 0 2 E…G†, then x2 y 0 u3 is a shortest path from H1 to H H1 . So 0 y 2 V2 …G†nV …H †. Hence by (1), (2), (3) and (4), H 0 ˆ H ‡ E…u3 yx2 y 0 u3 † is a subgraph of G which satis®es both (i) and (ii), but it contains more vertices in V2 …G† than H does, a contradiction. If dG …u1 ; y 0 †  2, then let p…u1 ; y 0 † be a shortest path from u1 to y 0 in G. Hence H 0 ˆ H ‡ …E…p† [ fus y 0 g [ E…p…u1 ; y 0 †††

……fus y 0 g [ E…p…u1 ; y 0 ††† \ E…H ††

is a subgraph of G which satis®es both (i) and (ii) but it contains more vertices in V2 …G† than H does, a contradiction. This completes the proof of Claim 1. ( Claim 2. jE…b†j  dia…G†

1 for any b 2 B1 …G†:

Proof of Claim 2. Otherwise there exists a branch b 2 B1 …G† with jE…b†j  dia…G†  3: Let x be the end of b with dG …x†  3 and y be the end of b with dG …y† ˆ 1: Then there exists a vertex z 2 V …G†nV …b† such that xz 2 E…G†: Obviously dG …y; z† ˆ jE…b†j ‡ 1  dia…G† ‡ 1 which is a contradiction. ( Claim 3. jE…b†j  dia…G† for any b 2 B…G†nBG …H †. Proof of Claim 3. Otherwise there exists a branch b 2 B…G† with E…b† \ E…H † ˆ / such that jE…b†j  dia…G† ‡ 1. By Claim 2, b 2 = B1 …G†. Without loss of generality, we assume that u and v be two endvertices of b, let p be a shortest path between u and v in G. Then jE…p†j  dia…G†  jE…b†j 1. Hence H 0 ˆ H ‡ …E…b† [ E…p††

…E…p† \ E…H ††

is a subgraph of G which satis®es both (i) and (ii) but it contains more vertices in V2 …G† than H does, a contradiction. This completes the proof of Claim 3. ( By Claims 1, 2 and 3, H 2 Edia…G† 1 …G†. Hence by Theorem 1.2, ham…G†  dia…G† 1. For any integer d  3 we construct a graph Gd to show that the upper bound is sharp. Let Ks …s  3† and Kt …t  3† be two complete graphs, and pd be a path of length d 2 such that Ks , Kt and pd are three edge-disjoint graphs. Now Gd is obtained by identifying two endvertices of pd with one vertex of Ks and one of Kt respectively. By Theorem 1.2, ham…Gd †  dia…Gd † 1 ˆ d 1 since the edgeless graph H with V …H † ˆ V …Ks † [ V …Kt † such that H 2 Ed 1 …Gd †. One can easily see that for each i 2 f0; 1; . . . ; d 2g, Li …Gd † is not hamiltonian and thus ham…Gd †  d 1 ˆ dia…Gd † 1. Hence ham…Gd † ˆ dia…Gd † 1 which completes the proof of Theorem 2.4. (

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Note 1. Gd constructed in this way shows that the upper bound dia…Gd † 1 ˆ d 1 for the hamiltonian index of a graph Gd is superior to the upper bound n D…G† ˆ s ‡ t ‡ d 3 maxfs; tg ˆ minfs; tg ‡ d 3: Note 2. In [1], it was claimed that Theorem 2.1 is best possible. Our result shows the falsity of this claim.

3. The Hamiltonian Index of the Complement of a Graph In this section, we consider the hamiltonian index of the complement of a graph. Before presenting our results, we need to state some useful results.  denote Theorem 3.1 [5]. Let G be a simple graph with at least 61 vertices, and let G the complement of G. One of the following holds.    

 is supereulerian. Either G or G  have a vertex of degree 1. Both G and G  is contractible to a K2;t for some odd integer t  3, and the Either G or G other one has either one or two vertices of degree 1.  is contractible to a K1;p for some integer p  1, and the other Either G or G one has exactly one isolated vertex.

Theorem 3.2 [3]. If G is a simple graph of order n  3 and d…G†  n=2; then G is a hamiltonian, i.e., ham…G† ˆ 0. Theorem 3.3 [8]. Let G be a connected simple graph other than a path. Then ham…G†  maxfjE…b†j : b 2 B…G†nfb 2 B…G† : G‰V …b†Š is a cycle of Ggg ‡ 1. Our main results are the following.  be two connected simple graphs of order n  61 other Theorem 3.4. Let G and G than paths. Then  1 (a) minfham…G†; ham…G†g

…5†

and the bound is sharp.  ˆ 1; then (b) If minfham…G†; ham…G†g   …n 1†=2 maxfham…G†; ham…G†g

…6†

 is isomorphic to the graph Ft and the equality holds if and only if one of G and G obtained by identifying one endvertex of a path Pt of length t 1 with exactly one vertex of a complete graph Kt of order t, where jV …Ft †j ˆ n ˆ 2t 1  61:

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 either is supereulerian or has one Proof. (a) By Theorem 3.1, one of G and G  is supereulerian, then by Theorem 1.1 its vertex of degree 1. If one of G and G hamiltonian index is at most one. Hence (5) holds.  has one vertex of degree 1. It remains to consider the case that one of G and G Without loss of generality, we assume that G has one vertex v with dG …v† ˆ 1 and  is connected, dG …u†  n 2: uv 2 E…G†: Since G Hence we distinguish the following two cases. Case 1. dG …u†  n

3:

We can take two vertices x; y in V …G†nfu; vg such that xu; yu 2 = E…G†; i.e.,  Let C be a circuit of G  with fvx; xu; uy; yvg  E…C† such that C xu; yu 2 E…G†.  as possible. Then dG…e; C† ˆ 0 for any e 2 E…G†.  contains as many edges in E…G†  has an edge e0 ˆ u1 u2 such that dG…e0 ; C† ˆ 1. Since Otherwise G  C 0 ˆ C ‡ fvu1 ; vu2 ; u1 u2 g is also a circuit such that fvu1 ; vu2 g  E…G†; fvx; xu; uy; yvg  E…C 0 † but C 0 contains more edges than C does. Thus dG…e; C† ˆ 0  Hence by Theorem 1.1, ham…G†   1 which implies that (5) for any e 2 E…G†. holds. Case 2. dG …u† ˆ n

2.

Let w be the vertex with uw 2 = E…G†: We consider the following three subcases. Subcase 2.1. dG …w†  n

5:

 i.e., We can take two vertices x; y in V …G†nfu; v; wg such that wx; wy 2 = E…G†,  with fwx; xv; vy; ywg  E…C† such that C wx; wy 2 E…G†: Let C be a circuit of G  as possible. Then by an argument similar to the contains as many edges in E…G†  Hence by Theorem 1.1, one in the proof of Case 1, dG…e; C† ˆ 0 for any e 2 E…G†:   1 which implies that (5) holds. ham…G† Subcase 2.2. dG …w† ˆ n

4.

Let x be the vertex in V …G†nfu; v; wg with xw 2 = E…G†: Hence tu; tw 2 E…G† for any t 2 V …G†nfu; v; w; xg. Let C be a circuit of G such that C contains as many edges in E…G† as possible. One can easily see that dG …e; C† ˆ 0 for any e 2 E…G†. Hence by Theorem 1.1, ham…G†  1 which implies that (5) holds. Subcase 2.3. dG …w† ˆ n

3:

Hence xu; xw 2 E…G† for any x 2 V …G†nfu; v; wg: By an argument similar to the one in the proof of subcase 2.2, ham…G†  1. This implies that (5) holds. The sharpness of (5) will be shown in the proof of (b).  ˆ1; say, ham…G†  ˆ minfham…G†; ham…G†g  ˆ 1; then (b) If minfham…G†; ham…G†g D…G† > n=2  ˆn Otherwise d…G† tradiction.

1

1

…7†

 ˆ 0, a conD…G†  n=2, by Theorem 3.2, ham…G†

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Set k…G† ˆ maxfjE…b†j : b 2 B…G†nfb 2 B…G† : G‰V …b†Š is a cycle of Ggg Let b 2 B…G†nfb 2 B…G† : G‰V …b†Š is a cycle of Gg such that jE…b†j ˆ k…G†. Now we distinguish the following two cases. Case 1. k…G†  …n

4†=2:

By Theorem 3.3, ham…G†  k…G† ‡ 1  …n Case 2. k…G†  …n

2†=2: Hence (6) holds.

3†=2:

Before our proceeding, we prove the following two claims. Claim 1. If b 2 B1 …G†, then k…G†  …n

1†=2.

Proof of Claim 1. Otherwise k…G†  n=2: It is easily seen that D…G†  n k…G†  n=2: By (7), k…G†  n D…G† < n …n=2 1†  …n ‡ 2†=2 and D…G†  …n 1†=2: Thus k…G†  …n ‡ 1†=2 and D…G† ‡ k…G† ˆ n: Without loss of generality, we assume that b ˆ xuk…G† uk…G† 1


u2 u1 with dG …u1 † ˆ 1 and V …G†nV …b† ˆ fv1 ; v2 ; . . . ; vD…G† 1 g; then  Cˆ

u1 xu2 v1 u3 v2


uk…G† vD…G† 1 u1 ; u1 xu2 v1 u3 v2


vD…G† 1 uk…G† u1 ;

if D…G† ˆ n=2 if D…G† ˆ …n 1†=2

 i.e., ham…G†  ˆ 0, a contradiction. is a hamiltonian cycle of G,

(

Claim 2. If b 2 B…G†nB1 …G†; then G‰E…G†nE…b†Š has only one nontrivial component. Proof of Claim 2. Otherwise G‰E…G†nE…b†Š has two nontrivial components. If k…G†  …n 2†=2, then D…G†  n …k…G† ‡ 2†  n ……n 2†=2 ‡ 2† ˆ n=2 1, which contradicts (7). If …n 3†=2  k…G† < …n 2†=2, i.e., k…G† ˆ …n 3†=2, then let G1 and G2 be the two nontrivial components of G‰E…G†nE…b†Š. Without loss of generality, we may assume that s ˆ jV …G1 †j  jV …G2 †j ˆ …n ‡ 5†=2 s, V …G1 † ˆ fu1 ; u2 ; . . . ; us g; b ˆ us us‡1


us‡…n 3†=2 and V …G2 † ˆ fus‡…n 3†=2 ; us‡…n 1†= 2 ; . . . ; un g. Hence 3  s  …n ‡ 5†=4, and thus u…n‡1†=2 2 V …b†n…V …G1 †[  Hence V …G2 ††. So fun u…n‡1†=2 ; u…n‡1†=2 u1 g  E…G†: C ˆ u1 u…n‡3†=2 u2 u…n‡5†=2


u…n

1†=2 un u…n‡1†=2 u1

 i.e., ham…G†  ˆ 0, a contradiction which completes is a hamiltonian cycle of …G†, the proof of Claim 2. ( Next we will distinguish two subcases. Subcase 2.1. b 2 B1 …G†: By Claim 1, k…G†  …n 1†=2: By (7) and …n 3†=2  k…G†  …n 1†=2, the SD…G† edgeless graph H 0 with V …H 0 † ˆ iˆ3 Vi …G† is a subgraph in Ek…G† …G†: So by Theorem 1.2, ham…G†  k…G†  …n 1†=2, i.e., (6) holds.

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Subcase 2.2. b 2 = B1 …G†; i.e., b 2 B…G†nB1 …G†. By Claim 2, G‰E…G†nE…b†Š has exactly one nontrivial component, let b ˆ u1 u2


uk…G†‡1 2 B…G†nB1 …G†: Then there exists a longest path p ˆ p…u1 ; uk…G†‡1 † from u1 to uk…G†‡1 in G‰E…G†nE…b†Š. Let H 0 be a subgraph of G with SD…G† V …H 0 † ˆ V …b† [ V …p† [ … iˆ3 Vi …G†† and E…H 0 † ˆ E…b† [ E…p†: By (7) and k…G†  …n 3†=2; 1 -0 [ V …G† @ A N …v†  n ……k…G† 2† ‡ D…G† ‡ 1†  3: v2V …H 0 † Hence H 0 is a subgraph in E7 …G†. Hence by Theorem 1.2, ham…G†  7 which implies that (6) holds by n  61. Clearly, Ft is a graph of order n ˆ 2t 1. The subgraph H of Ft , for which V …H † ˆ V …Kt † and E…H † ˆ /, is an element of Et 1 …Ft †. By Theorem 1.2, ham…Ft †  t 1 ˆ …n 1†=2: One can easily see that for each i 2 f0; 1; 2; . . . ; t 2g; Li …Ft † is not hamiltonian and thus ham…Ft †  t 1 ˆ …n 1†=2: Hence ham…FtT † ˆ …n 1†=2: Since the number of components of Ft …V …Pt †n …V …Pt † V …Kt ††† is greater than jV …Pt †j 1; Ft is not hamiltonian. Hence ham…Ft †  1: By (5), ham…Ft † ˆ 1. This also shows the sharpness of (5).  ˆ 1 and maxfham…G†; ham…G†g  ˆ On the other hand, if minfham…G†; hamGg  …n 1†=2; say, ham…G† ˆ …n 1†=2 and ham…G† ˆ 1 then, by the proof of (6), k…G† ˆ …n 1†=2 and G has a branch b 2 B1 …G† of length k…G†. Let G1 be the unique nontrivial component of G‰E…G†nE…b†Š: Then G1 is isomorphic to a complete graph of order …n ‡ 1†=2. Otherwise G1 has two vertices u1 and u2 such  Without loss of generality, we let V …G1 † ˆ that u1 u2 2 = E…G1 †; i.e., u1 u2 2 E…G†. fu1 ; u2 ; . . . ; u…n 1†=2 ; u…n‡1†=2 g and b ˆ ui u…n‡3†=2 u…n‡5†=2


un where ui 2 V …G1 †. If ui 2 fu1 ; u2 g, say, ui ˆ u1 , then C ˆ un u1 u2 un 1 u3 un 2 u4


u…n

1†=2 u…n‡3†=2 u…n‡1†=2 un

 which contradicts ham…G†  ˆ 1. is a hamiltonian cycle of G If ui 2 fu3 ; u4 ; . . . ; u…n‡1†=2 g, say, ui ˆ u3 , then C ˆ un u3 un 1 u1 u2 un 2 u4 u n 3 u5 un

4




u…n

1†=2 u…n‡3†=2 u…n‡1†=2 un

 which contradicts ham…G†  ˆ 1. This shows that G1 is is a hamiltonian cycle of G, isomorphic to a complete graph of order …n ‡ 1†=2. Hence G is isomorphic to Ft where t ˆ …n ‡ 1†=2. The proof of Theorem 3.4 is completed. ( By Theorem 3.4, we can obtain the following Nordhaus-Gaddum's inequalities.  are two connected simple graphs of order n  61 other Theorem 3.5. If G and G than paths and cycles, then

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(c) If D…G†  n=2

1, then

 n 0  ham…G† ‡ ham…G†

D…G†  n

3

 ˆ0 ham…G†
ham…G† (d) If D…G† > n=2

1, then  n 0  ham…G† ‡ ham…G†   …n 0  ham…G†
ham…G†

3 1†=2

and all bounds are sharp.  ˆ n 1 D…G†  n=2. By Theorem 3.2, Proof. (c) If D…G†  n=2 1; then d…G†  ham…G† ˆ 0. (c) follows by Theorem 2.2. Clearly, (c) is sharp by Theorem 2.2. (d) It follows from Theorem 2.2 and the proof of Theorem 3.4. One can easily see that the lower bounds of (d) are sharp. The sharpness of upper bounds of (d) follows from Theorem 2.2 and Theorem 3.4. ( Acknowledgment. The author would like to thank one of the anonymous referees

for the careful corrections of my manuscript. Also thanks another referee for the information on NebeskyÂ's results [2]. These comments made me improve Theorem 3.5 to Theorem 3.4. References

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Received: July 17, 1998 Final version received: September 13, 1999