JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 17, 535-548 (2001) A Family of Trivalent 1-Hamiltonian Graphs With Diameter O(log n) JENG-JUNG WANG, TING-YI SUNG* AND LIH-HSING HSU** Department of Information Engineering I- Shou University Kaohsiung, Taiwan 840, R.O.C. *Institute of Information Science Academia Sinica Taipei, Taiwan 115, R.O.C. **Department of Computer and Information Science National Chiao Tung University Hsinchu, Taiwan 300, R.O.C. E-mail: [email protected] In this paper, we construct a family of graphs denoted by Eye(s) that are 3-regular, 3-connected, planar, hamiltonian, edge hamiltonian, and also minimal 1-hamiltonian. Furthermore, the diameter of Eye(s)isO(log n), where n is the number of vertices in the graph and to be precise, n =6(2s − 1) vertices. Keywords: hamiltonian, edge hamiltonian, 1-vertex hamiltonian, 1-edge hamiltonian, 1-hamiltonian, diameter, Moore bound 1. INTRODUCTION Given a graph G =(V, E), V(G)=V and E(G)=E denote the vertex set and the edge set of G, respectively. All graphs considered in this paper are undirected graphs. A sim- ple path (or path for short) is a sequence of adjacent edges (v1, v2), (v2, v3), …, (vm-2, vm-1), (vm-1, vm), written as 〈v1, v2, v3,…,vm〉, in which all of the vertices v1, v2, v3,…,vm are distinct except possibly v1 = vm. The path 〈v1, v2, v3,…,vm〉 is also called a cycle if v1 = vm and m ≥ 3. A cycle that traverses every vertex in the graph exactly once is called a ham- iltonian cycle. A graph that contains a hamiltonian cycle is called a hamiltonian graph or said to be hamiltonian.Agraphisedge hamiltonian if each edge in the graph is incident with some hamiltonian cycle in the graph. The diameter of graph G is the maximum dis- tance among all pairs of vertices in G,wheredistance means the length of a shortest path joining two distinct vertices in G. For V' ⊂ V and E' ⊂ E, G − V' − E' denotes the graph obtained by removing all of the vertices in V' from V and removing the edges incident with at least one vertex in V' and also all of the edges in E' from E.Letk be a positive integer. A graph G is k-hamiltonian if G − V' − E' is hamiltonian for any set V' ⊂ V and E' ⊂ E with |V'| + |E'|≤ Received April 30, 1999; revised December 17, 1999; accepted February 24, 2000. Communicated by Jang-Ping Sheu. 535 536 JENG-JUNG WANG, TING-YI SUNG AND LIH-HSING HSU k. It is clear that every k-hamiltonian graph has at least k + 3 vertices. Moreover, the de- gree of every vertex in a k-hamiltonian graph is at least k +2.Ak-hamiltonian graph having n vertices is said to be minimal if it contains the least number of edges among all k-hamiltonian graphs having n vertices. A graph is k-connected if the subgraph induced by G − V' is connected for any subset V' of V with |V'|≤k − 1. A graph is r-regular if the number of neighbors of each vertex is r. Mukhopadhyaya and Sinha [6] proposed a family of 1-hamiltonian graphs which n are also planar. These graphs are hamiltonian and have diameter of + 2 if n is n 6 even, and + 3 if n is odd. Harary and Hayes also studied similar problems on 8 graphs, called k-edge hamiltonian and k-vertex hamiltonian graphs in [3] and [4], respec- tively. A graph G is k-edge hamiltonian if G − E' is hamiltonian for any E' ⊂ E with |E' | = k; G* is said to be minimal k-edge hamiltonian if G* contains the least number of edges among all k-edge hamiltonian graphs having the same number of vertices as G*. A graph G is k-vertex hamiltonian if G − V' is hamiltonian for any V' ⊂ V with |V'|=k; G* is said to be minimal k-vertex hamiltonian if G* contains the least number of edges among all k-vertex hamiltonian graphs having the same number of vertices as G*. For any positive integer k, Harary and Hayes presented families of minimal k-edge hamiltonian graphs and minimal k-vertex hamiltonian graphs in [3] and [4], respectively. In particular, the family of minimal 1-edge hamiltonian graphs proposed in [3] are identical to the family of minimal 1-vertex hamiltonian graphs proposed in [4]. Hence, this family of graphs are 1-hamiltonian. Furthermore, each graph is planar, hamiltonian, and of diameter n + 2 n n +1 n + (( +1) mod 2) if n is even, or + (( +1) mod 2) if n is odd, where 4 2 4 2 n is the number of vertices in the graph. Harary and Hayes questioned in [4] whether their proposed minimal 1-vertex ham- iltonian graphs are the only such graphs, and asked for characterization of all minimal 1-vertex hamiltonian graphs. The family of graphs proposed by Mukhopadhyaya and Sinha mentioned above are also minimal 1-vertex hamiltonian and provide a counterex- ample to their question. Recently, Wang, Hung, and Hsu [9] presented another family of 1-hamiltonian graphs, each of which is planar, hamiltonian, 3-regular, and of diameter O( n) with n vertices in the graph. On the other hand, we can find minimal 1-vertex hamiltonian graphs which are non-hamiltonian. A hypohamiltonian graphisa non-hamiltonian 1-vertex hamiltonian graph [8]. Hypohamiltonian graphs have been extensively discussed in the literature [5], [7] and [8]. The Petersen graph and the coxeter graph are famous examples of hypohamiltonian graphs [7]. Characterization of all hypo- hamiltonian graphs is a difficult problem in graph theory. We are interested in finding more families of minimal 1-hamiltonian graphs that are also hamiltonian. The three families of minimal 1-hamiltonian graphs presented in [4], [6] and [9] are planar, 3-regular and hamiltonian. It is natural to ask whether we can find such graphs with smaller diameter. This problem relates to the famous (n, d, D) problem in which we want to construct a graph of n vertices with maximum degree d such that the diameter D is minimized. When d and n are given, the lower bound on diameter D, called the Moore 2 bound (on diameter), is given by D ≥ log − n − [2]. In this paper, we propose a fam- d 1 d TRIVALENT 1-HAMILTONIAN GRAPHS WITH DIAMETER O(LOG n) 537 ily of minimal 1-hamiltonian graphs that are hamiltonian, edge hamiltonian, planar, 3-regular and 3-connected. Furthermore, the diameter of our graphs is no more than n + 6 4log , i.e., less than 4 times of the Moore bound. Since all graphs in our family 2 6 are 3-regular, they are minimal 1-hamiltonian graphs. 2. DEFINITIONS k-1 Let N0 =3andNk =9⋅2 for any positive integer k.Let[i]m denote i mod m.Aneye graph Eye(s), s ≥ 1, is a graph with s + 1 layers of concentric cycles; see Fig. 1 for an illustration. These s + 1 cycles are denoted by I0, I1, I2,…,Is-1,andOs. In particular, Os is Us−1 ∪ the outermost cycle. The vertex set V(Eye(s)) is given by k =0V (I k ) V (Os ) where Fig. 1. Examples of eye graphs. V(Ik)={(k, j) | 0 ≤ j ≤ Nk − 1} for 0 ≤ k ≤ s − 1and V(Os)={(s, j) | 0 ≤ j ≤ Ns − 1and[j]3 =0}. For vertex (k, j), k and j are referred to as the first and the second coordinate, re- spectively. Throughout this paper all computations on the second coordinate of a vertex at the kth concentric cycle are carried out with modulo Nk. The graph Eyes(s) contains two types of edges, i.e., cycle edges, denoted by ek,j,andintercycle edges, denoted by k +1 ek, j , which are given as follows: ((k, j),(k, j + 1)) for 0 ≤ k ≤ s −1 and 0 ≤ j ≤ N −1, e = k k , j + = ≤ ≤ − ≠ ((k, j),(k, j 3)) for k s, 0 j N s 1 and [ j]3 0. ((0, j), (1,3 j)) for k = 0 and j = 0,1,2 e k +1 = k , j + + ≤ ≤ − ≤ ≤ − ≠ ((k, j), (k 1,2 j [ j]3 )) for 1 k s 1, 0 j N k 1 and [ j]3 0. The set {ek,j |0≤ j ≤ Nk − 1} is denoted by E(Ik)if0≤ k ≤ s − 1, and denoted by E(Os) k +1 k +1 ≤ ≤ − if k = s. We use Ek to denote the set {ek, j 0 j N k 1}. The edge set of 538 JENG-JUNG WANG, TING-YI SUNG AND LIH-HSING HSU + = Us−1 ∪ ∪ Us−1 k 1 Eye(s) is defined asE(Eye(s)) k =0 E(I k ) E(O s ) ( k =0 E k ). The graph Eye(s) − + s−1 k −1 + ⋅ s 1 = s − s − is 3-regular and contains 3 9∑k =1 2 3 2 6(2 1) vertices and 9(2 1) edges. On the other hand, graphs Eye(s +1),s ≥ 1, can be recursively drawn (or con- structed) from Eye(s) by performing the following two steps: SUBDIVISION. Subdivide each edge es,j of Os,0≤ j ≤ Ns − 1and[j]3 = 0, into a path of length 3; i.e., replace es,j with a path 〈(s, j), (s, j +1),(s, j +2),(s, j +3)〉 to connect its two ends (s, j)and(s, j + 3). Rename Os as Is. EXTENSION. Construct graph Os+1 as a concentric cycle outside Is and join every vertex (s, j)inIs with vertex (s +1,2j +[j]3)inOs+1 for 1 ≤ j ≤ Ns − 1and[j]3 ≠ 0.