On the Extremal Number of Edges in Hamiltonian Graphs
* TUNG-YANG HO+, CHENG-KUAN LIN1, JIMMY J. M. TAN1, D. FRANK HSU2AND LIH-HSING HSU3
+
Department of Information Management Ta Hwa Institute of Technology
Hsinchu, 307 Taiwan E-mail: hoho@thit.edu.tw
1
Department of Computer Science National Chiao Tung University
Hsinchu, 300 Taiwan
2
Department of Computer and Information Science Fordham University
New York, NY 10023, U.S.A.
3
Department of Computer Science and Information Engineering Providence University
Taichung, 433 Taiwan
Assume that n and δ are positive integers with 2 ≤ δ < n. Let h(n, δ) be the minimum number of edges required to guarantee an n-vertex graph with minimum degree δ(G) ≥ δ to be hamiltonian, i.e., any n-vertex graph G with δ(G) ≥ δ is hamiltonian if |E(G)| ≥ h(n, δ). We prove that h(n, δ) = C(n − δ, 2) + δ2 + 1 if 1 3 (( 1) mod 2) ,
6 n n δ ≤⎢⎣ + + × + ⎥⎦ h(n, δ) = C(n − ⎢n2−1⎥, 2)+⎢n2−1⎥2+1 ⎣ ⎦ ⎣ ⎦ if ⎢n + + ×1 3 (( 6n+1) mod 2)⎥< ≤δ ⎢n2− 1 ,⎥ ⎣ ⎦ ⎣ ⎦ and h(n, δ) =⎡ ⎤⎢ ⎥n2δ ifδ>⎣⎢n 1 .2− ⎥⎦
Keywords: complete graph, cycle, hamiltonian, hamiltonian cycle, edge-fault tolerant ham-iltonian
1. INTRODUCTION
Throughout this paper, we use C(a, b) to denote the number of combinations of “a” numbers taking “b” numbers at a time, where a, b are positive integers and a ≥ b. For the graph definitions and notations, we follow [1]. Let G = (V, E) be a simple graph if V is a finite set and E is a subset of {(u, v) | (u, v) is an unordered pair of V}. We say that V is the vertex set and E is the edge set. Two vertices u and v are adjacent if (u, v) ∈ E. The com-plete graph Kn is the graph with n vertices such that any two distinct vertices are adjacent.
The degree of a vertex u in G, denoted by degG(u), is the number of vertices adjacent to
u. We use δ(G) to denote min{degG(u)| u ∈ V(G)}. We use c(G) to denote the number of
connected components in G. A path, 〈v0, v1, …, vm-1〉, is an ordered list of distinct vertices
such that vi and vi+1 are adjacent for 0 ≤ i ≤ m − 2. A cycle is a path with at least three
vertices such that the first vertex is the same as the last one. A hamiltonian cycle of G is a Received December 11, 2009; revised April 6, 2010; accepted April 23, 2010.
Communicated by Ding-Zhu Du.
cycle that traverses every vertex of G exactly once. A graph is hamiltonian if it has a ham-iltonian cycle.
In the past years, the studies on hamiltonian graphs have largely focused on their re-lationship to the Four Color Problem. More recently, the study of hamiltonian cycle in gen-eral graphs has been fueled by practical applications and by the issue of complexity. No easily testable characterization is known for hamiltonian graphs. Some sufficient condi-tions have been investigated. Two milestones of these sufficient condicondi-tions are obtained by Ore and Dirac. Ore [9] proved that any n-vertex graph with at least C(n, 2) − (n − 3) edges is hamiltonian, and there exists an n-vertex non-hamiltonian graph with C(n, 2) − (n − 2) edges. Dirac [4] obtains the following sufficient condition based on the minimum degree. Theorem 1 Let G be an n-vertex graph with n ≥ 3 andδ( )G ≥2n. Then G is hamilto- nian. Moreover, there exists an n-vertex non-hamiltonian graph G with δ( )G <2n.
Erdős [5] presents the following sufficient condition based on the combination of the number of edges and the minimum degree.
Theorem 2 Let G be an n-vertex graph with n ≥ 6δ(G). Then G is hamiltonian if |E(G)| >
C(n − δ(G), 2) + δ(G)2.
In this paper, we consider a result in a setting more general than Theorem 2. Our re-sult (Theorem A) will also include Theorem 1 as a special case. Since a graph G with δ(G) = 1 is not hamiltonian, we consider graph G with δ(G) ≥ 2 in the following. Assume that n and δ are positive integers with 2 ≤ δ < n. Let h(n, δ) be the minimum number of edges required to guarantee an n-vertex graph with δ(G) ≥ δ to be hamiltonian. So any n-vertex graph G with δ(G) ≥ δ is hamiltonian if |E(G)| ≥ h(n, δ). We will prove the following theo-rem.
Theorem A Assume that n and δ are positive integers with 2 ≤ δ < n. Then
2 2 1 3 (( 1) mod 2) 6 1 3 (( 1) mod 2) 1 1 1 2 2 6 2 1 2 2 ( , 2) 1 if , ( , ) ( , 2) 1 if , if . n n n n n n n n n C n h n C n δ δ δ δ δ δ δ + + × + + + × + − − − − ⎧ − + + ≤ ⎢⎢ ⎥ ⎥ ⎪ ⎣ ⎦ ⎪⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎨ = −⎣ ⎦ +⎣ ⎦ + ⎢ ⎥< ≤⎣ ⎦ ⎣ ⎦ ⎪ ⎪⎡ ⎤ >⎢ ⎥ ⎪⎢ ⎥ ⎣ ⎦ ⎩
Roughly speaking, h(n, δ) depends on n and δ when δ ≤6n or
2,
n
δ > h(n, δ) de- pends only on n when 6n< ≤δ 2n. The latter is our main contribution. We use an example to illustrate Theorem A with the case that n = 16.
n 16 16 16 16 16 16 16 16 16 16 16 16 16 16
δ 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Any 16-vertex graph G with δ(G) ≥ 2 is hamiltonian if |E(G)| ≥ 96, with δ(G) ≥ 3 is hamiltonian if |E(G)| ≥ 88, with δ(G) ≥ 4, …, 7 is hamiltonian if |E(G)| ≥ 86, with δ(G) ≥ 8 is hamiltonian if |E(G)| ≥ 64, with δ(G) ≥ 9 is hamiltonian if |E(G)| ≥ 72, with δ(G) ≥ 15 is hamiltonian if |E(G)| ≥ 120.
We compare our result Theorem A with Theorems 1 and 2, and make some remarks here. Theorem 1 considers the case δ ≥2n, which is the same as the case 1
2 .
n
δ > ⎣⎢ − ⎥⎦ And
Theorem 2 discusses the case δ ≤6n.We notice that there are still cases for
6 n < δ ≤ 1 2 n− ⎢ ⎥ ⎣ ⎦
left open. So our result fills up the gap and unifies the previous results.
We defer the proof of Theorem A to section 4. We first give an application of Theo-rem A, which is the original motivation of this paper. In particular, we establish the rela-tionship between h(n, g) and g-conditional edge-fault tolerant hamiltonicity of the com-plete graph Kn. Then we give some preliminary results in section 3. Finally, section 4 gives
the proof of Theorem A.
2. APPLICATIONS
A hamiltonian graph G is k edge-fault tolerant hamiltonian if G − F remains hamil-tonian for every F ⊂ E(G) with |F| ≤ k. The edge-fault tolerant hamiltonicity, He(G), is
defined as the maximum integer k such that G is k edge-fault tolerant hamiltonian if G is hamiltonian and is undefined otherwise. Assume that G is a hamiltonian graph and x is a vertex such that degG(x) = δ(G). We arbitrary choose degG(x) − 1 edges from those edges
incident to x to form an edge faulty set F. Obviously, degG-F(x) = 1 and hence, G − F is not
hamiltonian. Therefore, He(G) ≤ δ(G) − 2 if He(G) is defined. It is easy to check that He(Kn)
= n − 3 for n ≥ 3. In Latifi et al. [7], it is proved that He(Qn) = n − 2 for n ≥ 2 where Qn is
the n-dimensional hypercube. In Li et al. [8], it is proved that He(Sn) = n − 3 for n ≥ 3
where Sn is the n-dimensional star graph.
Chan and Lee [2] began the study of the existence of hamiltonian cycle in a graph such that each vertex is incident to at least g nonfaulty edges. A graph G is g-conditional k edge-fault tolerant hamiltonian if G − F is hamiltonian for every F ⊂ E(G) with |F| ≤ k and minimum degree δ(G − F) ≥ g. The g-conditional edge-fault tolerant hamiltonicity, He
g(G), is defined as the maximum integer k such that G is g-conditional k edge-fault
tol-erant hamiltonian if G is hamiltonian and is undefined otherwise. Chan and Lee [2] proved that He
g(Q
n) ≤ 2g-1(n − g) − 1 for n > g ≥ 2 and the equality holds for g = 2.
Fu [6] studied the 2-conditional edge-fault tolerant hamiltonicity of the complete graph. The following result is in [6]:
Suppose F ⊂ E(Kn) and δ(Kn − F) ≥ 2, where n ≥ 4. If n ∉ {7, 9} (respectively, n ∈
{7, 9}) then Kn − F is hamiltonian, where |F| ≤ 2n − 8 (respectively, |F| ≤ 2n − 9).
In the conclusion of [6], it is claimed that the above statement is optimal. We restate this result using our terminology.
He2(Kn) = 2n − 8 for n ∉ {7, 9} and n ≥ 4, He2(K7) = 5, and He2(K9) = 9.
Now, we extend the result in [6] and use our main result Theorem A to compute He g (Kn) for any 1 ≤ g < n. Theorem 3 He g (Kn) = C(n, 2) − h(n, g) for any 1 ≤ g < n.
Proof: Let F be any faulty edge set of Kn with |F| ≤ C(n, 2) − h(n, g) such that δ(Kn − F) ≥
g. Obviously, |E(Kn − F)| ≥ h(n, g). By Theorem A, Kn − F is hamiltonian. Thus, He g
(Kn) ≥
C(n, 2) − h(n, g).
Now, we prove that He g
(Kn) ≤ C(n, 2) − h(n, g). Assume that He g
(Kn) ≥ C(n, 2) − h(n, g)
+ 1. Let G be any graph with h(n, g) − 1 edges such that δ(G) ≥ g. Let F be E(Kn) − E(G).
In other words, G = Kn − F. Obviously, |F| = C(n, 2) − h(n, g) + 1. Since He g
(Kn) ≥ C(n, 2)
− h(n, g) + 1, G is hamiltonian. This contradicts to the definition of h(n, g). Thus, He g (Kn) ≤ C(n, 2) − h(n, g). Therefore, He g (Kn) = C(n, 2) − h(n, g) for any 1 ≤ g < n. 3. PRELIMINARY RESULTS
Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs. The union of G1 and G2, G1 + G2, has edge set E1 ∪ E2 and vertex set V1 ∪ V2 with V1 ∩ V2 = φ. The join of G1 and G2, G1 ∨ G2, is obtained from G1 + G2 by joining each vertex of G1 to each vertex of G2.
For 1 ≤ m < n/2, let Cm,n be the graph (Km + Kn-2m) ∨ Km and S be the set of vertices
in Km; i.e., Cm,n − S =Km + Kn-2m. We know that c(Cm,n − S) = m + 1 > |S|. Therefore, Cm,n is
not hamiltonian.
The degree sequence of an n-vertex graph is the list of vertices degree, in nonde-creasing order, as d1 ≤ d2 ≤ … ≤ dn. A sequence of real numbers (p1, p2, …, pn) is said to
be majorised by another sequence (q1, q2, …, qn) if pi ≤ qi for 1 ≤ i ≤ n. A graph G is
de-gree-majorised by a graph H if |V(G)| = |V(H)| and the nondecreasing degree sequence of G is majorised by that of H. For instance, the 5-cycle is degree-majorised by the com-plete bipartite graph K2,3 because (2, 2, 2, 2, 2) is majorised by (2, 2, 2, 3, 3).
Chvátal [3] points out that the family of degree-maximal non-hamiltonian graphs (those are not degree-majorised by others) are exactly Cm,n’s, i.e., any n-vertex non-ham-
iltonian graph is degree-majorised by some Cm,n.
Corollary 1 Let n ≥ 5. Assume that G is an n-vertex non-hamiltonian graph. Then δ(G) ≤ ⎢n 12− ⎥
⎣ ⎦ and |E(G)| ≤ 1
2
( ), ,
max{| (E Cδ G n)|, | (E C⎢⎣n−⎥⎦ n)|}.
Proof: Let G be any n-vertex non-hamiltonian graph. With Theorem 1, δ(G) ≤ ⎢⎣n 12− ⎥⎦. And we know that G is degree-majorised by Cm,n for some integer m. Since δ(Cm,n) = m,
δ(G) ≤ m ≤ 1
2 .
n−
⎢ ⎥
⎣ ⎦ Therefore, |E(G)| ≤ max{|E(Cm,n)| | δ(G) ≤ m ≤⎣⎢n 12− ⎥⎦}. Since |E(Cm,n)|
= 12(m2 + (n − 2m)(n − m − 1) + m(n − 1)) is a quadratic function with respect to m and the the maximum value of it occurs at the boundary m = δ(G) or m = 1
2 , n− ⎢ ⎥ ⎣ ⎦ |E(G)| ≤ max{|E (Cδ(G),n)|, 1 2 , | (E C⎢n−⎥n)|}. ⎣ ⎦ By Corollary 1, we have the following corollary.
Corollary 2 Assume that G is an n-vertex graph with n ≥ 5. Then G is hamiltonian if |E(G)| ≥ max{|E(Cδ(G),n)|, 1
2 , | (E C⎢n−⎥n)|}
⎣ ⎦ + 1.
Lemma 1 Assume that n and k are integers with n ≥ 5 and 1≤ ≤ ⎣k ⎢n2− 1⎥⎦. Then |E(Ck,n)|
≥ 1 2 , | (E C⎢n−⎥ n)| ⎣ ⎦ if and only if 1 3 (( 1)mod2) 6 1≤ ≤ ⎢k ⎢n+ + × n+ ⎥⎥ ⎣ ⎦ or k= ⎣⎢n2− 1⎥⎦.
Proof: We first prove the case that n is even. We claim that |E(Ck,n)| ≥
2 1,
| (E Cn− n)| if and only if 1 ≤ k ≤ ⎢n 46+ ⎥
⎣ ⎦ or k = n2 − 1. Suppose that |E(Ck,n)| <
2 1, | (E Cn− n)|. Then |E(Ck,n)| = 12 (k2 + (n − 2k)(n − k − 1) + k(n − 1)) < 2 2 1, 12 2 2 2 | (E Cn− n)|= ((n−1) +(n−2(n−1))(n−(n− −1) 2 1) (+ n−1)(n−1)). This implies 3k2 + (1 − 2n)k + 1 2 1 4 2 ( n + n−2) 0,< which means (k − n2 + 1)(3k − 2n− 2) < 0. Thus, |E(C k,n)| < 2 1,
| (E Cn− n)| if and only if n 4+6 < < −k n2 1. Note that n and k are integers with n being even, n ≥ 6, and 1 ≤ k ≤ 2n − 1. Thus, |E(Ck,n)| ≥
2 1, | (E Cn− n)| if and only if 1 ≤ k ≤⎢n 46+ ⎥
⎣ ⎦ or k = 2n − 1.
For odd integer n, using the same method, we can prove that |E(Ck,n)| <
2 1, | (E Cn− n)| if and only if n6+1< <k n2−1. Given that n ≥ 5, and 1
2 1≤ <k n− , then |E(C k,n)| ≥ 1 2, | (E Cn− n)| if and only if 1 k ⎢n 16+ ⎥
≤ ≤ ⎣ ⎦ or k=n2− 1. Therefore, the result follows.
4. PROOF OF THEOREM A
By brute force, we can check that h(3, 2) = 3, h(4, 2) = 4, and h(4, 3) = 6. Therefore, the theorem holds for n = 3, 4. Next, we consider the cases that 1 ≤ δ ≤ 1
2
n−
⎢ ⎥
⎣ ⎦ and n ≥ 5.
Suppose that 1 ≤ δ ≤ ⎢⎣⎢n + + ×1 3 (( 6n+1)mod2)⎥⎥⎦. By Lemma 1, |E(Cδ,n)| ≥ 1 2 , | (E C⎢n−⎥n)|.
⎣ ⎦ Let G be any n-vertex graph with δ(G) ≥ δ and |E(G)| ≥ |E(Cδ,n)| + 1 = C(n − δ, 2) + δ2 + 1. By Corollary 2, G is hamiltonian. Therefore, h(n, δ) ≤ C(n − δ, 2) + δ2 + 1. Since δ <
2,
n
Cδ,n is not hamiltonian. Thus, h(n, δ) > |E(Cδ,n)| = C(n − δ, 2) + δ2. Hence, h(n, δ) = C(n − δ, 2) + δ2 + 1.
Suppose that ⎢n + + ×1 3 (( 6n+1)mod2)⎥< ≤ ⎣δ ⎢n2− 1⎥. ⎦ ⎢ ⎥ ⎣ ⎦ By Lemma 1, |E(Cδ,n)| ≤ 1 2 , | (E C⎢n−⎥n)|. ⎣ ⎦ Let G be any n-vertex graph with δ(G) ≥ δ and |E(G)| ≥
2 1,
| (E Cn− n)| + 1 = C n( − −2n 1, 2)+ 2
(n − 1)2 + 1. By Corollary 2, G is hamiltonian. Therefore, h(n, δ) ≤
2 2
( n 1, 2) (n
C n− − + −
1)2 + 1. And we know that 2 1,
n n
C − is not hamiltonian. Thus, h(n, δ) > 2 1, | (E Cn− n)| = C(n − 2 n − 1, 2) + 2 (n − 1)2. Hence, h(n, δ) = 2 2 2 ( n 1, 2) (n 1) 1. C n− − + − +
Finally, we consider the case that δ > 1 2
n−
⎢ ⎥
⎣ ⎦ and n ≥ 5. Let G be any graph with δ(G)
≥ δ > 1
2 .
n−
⎢ ⎥
⎣ ⎦ By Theorem 1, G is hamiltonian. Obviously, |E(G)| ≥⎡⎢n2δ⎤⎥. Thus, h(n, δ) = 2 .
nδ
⎡ ⎤ ⎢ ⎥
REFERENCES
1. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North Holland, New York, 1980.
2. M. Y. Chan and S. J. Lee, “On the existence of hamiltonian circuits in faulty hyper-cubes,” SIAM Journal on Discrete Mathematics, Vol. 4, 1991, pp. 511-527.
3. V. Chvátal, “On Hamilton’s ideals,” Journal of Combinatorial Theory B, Vol. 12, 1972, pp. 163-168.
4. G. A. Dirac, “Some theorems on abstract graphs,” in Proceedings of London Mathe-matical Society, Vol. 2, 1952, pp. 69-81.
5. P. Erdős, “Remarks on a paper of Pósa,” Magyar Tudományos Akadémia Matematikai Kututató Intézeténk Közleményei, Vol. 7, 1962, pp. 227-229.
6. J. S. Fu, “Conditional fault hamiltonicity of the complete graph,” Information Proc-essing Letters, Vol. 107, 2008, pp. 110-113.
7. S. Latifi, S. Q. Zheng, and N. Bagherzadeh, “Optimal ring embedding in hypercubes with faulty links,” in Proceedings of IEEE Symposium on Fault-Tolerant Computing, 1992, pp. 178-184.
8. T. K. Li, J. J. M. Tan, and L. H. Hsu, “Hyper hamiltonian laceability on the edge fault star graph,” Information Sciences, Vol. 165, 2004, pp. 59-71.
9. O. Ore, “Note on hamilton circuits,” The American Mathematical Monthly, Vol. 67, 1960, pp. 55.
Tung-Yang Ho (何東洋) received the B.S. and M.S. degrees in Applied Mathematics in 1984 and 1989, respectively, and the Ph.D. degree in Computer Science in 1995, all from National Chiao Tung University, Taiwan, R.O.C. Currently, he is a Professor in the Department of Information Management, Ta Hwa Institute of Technology, Hsinchu, Taiwan, R.O.C. His research interests are applications of graph theory in network architectures and graph algorithms.
Cheng-Kuan Lin (林政寬) received his M.S. degree in Mathematics from National Central University, Taiwan, R.O.C. in 2002. His research interests include interconnection networks and graph theory.
Jimmy J. M. Tan (譚建民) received the B.S. and M.S. de-grees in Mathematics from the National Taiwan University in 1970 and 1973, respectively, and the Ph.D. degree from Carleton University, Ottawa, in 1981. He has been on the faculty of the Department of Computer Science, National Chiao Tung Univer-sity, since 1983. His research interests include design and analy-sis of algorithms, combinatorial optimization, and interconnec-tion networks.
D. Frank Hsu is the Clavius Distinguished Professor of Science and a Professor of Computer and Information Science at Fordham University in New York. He has been visiting profes-sor/scholar at Keio University (Tokyo, Japan), JAIST (Kanazawa, Japan), Taiwan University, TsingHua University (Hsinchu, Tai-wan), University of Paris-Sud and CNRS, MIT and Boston Uni-versity. Dr. Hsu’s research interests are combinatorics, algorithms, optimization, network interconnection and communication, ma-chine learning, data mining and information fusion, applied in-formatics in biomedicine, health, finance, business, neuroscience, and cybersecurity. Dr. Hsu has served on several editorial boards and currently is an as-sociate editor of Pattern Recognition Letter and Special Issue Editor for the Journal of Interconnection Networks (JOIN). He was Conference Co-Chair for the International Con-ference on Cyber Security (ICCS 2009) held in New York City, January 6-8, 2009, and for International Symposium on Pervasive Systems, Algorithms and Networks (I-SPAN 2009) held in Kaohsiung, Taiwan December 14-16, 2009. Dr. Hsu is a senior member of IEEE, a Foundation Fellow of the Institute of Combinatorics and Applications (ICA), a Fellow of the New York Academy of Sciences (NYAS) and the International Society of Intelligent Biological Medicine (ISIBM).
Lih-Hsing Hsu (徐力行) received his B.S. degree in Mathe-matics from Chung Yuan Christian University, Taiwan, R.O.C., in 1975, and his Ph.D. degree in Mathematics from the State Uni-versity of New York at Stony Brook in 1981. From 1981 to 1985, he was Associate Professor at Department of Applied Mathematics at National Chiao Tung University in Taiwan. In 1985, he was a Professor in National Chiao Tung University. After 1988, he joined with Department of Computer and Information Science of National Chiao Tung University. In 2004, he is retied from National Chiao Tung University by holding a title as honorary scholar of National Chiao Tung University. He is currently a Professor in the Department of Computer Sci-ence and Information Engineering, ProvidSci-ence University, Taichung, Taiwan, R.O.C. His research interests include interconnection networks, algorithms, graph theory, and VLSI layout.
