Construction Schemes of Hamiltonian Laceable and Bipancyclic Graphs

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(1)Construction Schemes of Hamiltonian Laceable and Bipancyclic Graphs ∗ Y-Chuang Chen, Jimmy J. M. Tan†, Tyne Liang, and Lih-Hsing Hsu Department of Computer and Information Science National Chiao Tung University Hsinchu, Taiwan 300, R.O.C.. Abstract In this paper, we study some hamiltonian properties of bipartite graphs. Every hamiltonian bipartite graph G = (V0 ∪ V1 , E) satisfies |V0 | = |V1 |. Since the colors of a path in bipartite graphs alternate, any hamiltonian bipartite graphs cannot be hamiltonian connected. Thus, the concepts of hamiltonian laceability, strongly hamiltonian laceability, hyper-hamiltonian laceability, bipancyclicity, and edge-bipancyclicity for hamiltonian bipartite graphs are attractive topics in interconnection networks. In this paper, we propose several methods, which extend the result in [7], to construct hamiltonian laceable, strongly hamiltonian laceable, and hyper-hamiltonian laceable graphs. We also show that our construction schemes preserve bipancyclic and edge-bipancyclic properties.. Keywords–Hamiltonian laceable, Strongly hamiltonian laceable, Hyper-hamiltonian laceable, Bipancyclic, Edge-bipancyclic.. 1. Introduction. For the graph definitions and notations we follow [2]. G = (V, E) is a graph if V is a finite set and E is a subset of {(a, b) | (a, b) is an unordered pair of V }. We say that V is the node set and E is the edge set. Two nodes a and b are adjacent if (a, b) ∈ E. A ∗. This work was supported in part by the National Science Council of the Republic of China under Contract NSC 90-2213-E-009-149. † Correspondence to: Professor Jimmy J. M. Tan, Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan 300, R.O.C. e-mail: [email protected].. 1.

(2) path is a sequence of adjacent nodes, written as v0 , v1 , v2 , · · · , vm , in which all the nodes v0 , v1 , v2 , · · · , vm are distinct except possibly v0 = vm . We also write the path v0 , P, vm where P = v0 , v1 , v2 , · · · , vm . A path is a hamiltonian path if its nodes are distinct and they span V . A cycle is a hamiltonian cycle if it traverses every node of G exactly once. A graph G is hamiltonian if it has a hamiltonian cycle; and G is hamiltonian connected if for any two nodes of G, there exists a hamiltonian path joining these two. A graph G = (V0 ∪ V1 , E) is bipartite if V (G) is the union of two disjoint sets V0 and V1 such that each edge consists of one node from each set. We say that V0 and V1 are two different colored node sets. Any hamiltonian bipartite graph G = (V0 ∪ V1 , E) satisfies |V0 | = |V1 |. We observe that the colors of a path in bipartite graphs alternate, any hamiltonian bipartite graph is not hamiltonian connected. Simmons [15] introduced the concept of hamiltonian laceability for hamiltonian bipartite graphs. A bipartite graph is equitable if it has the same number of nodes in each of its two colors. If the number of nodes in each of its two color sets differ by exactly one, it is called nearly equitable. A bipartite graph is defined [15] to be hamiltonian laceable if (a) it is equitable and whenever x and y are nodes of opposite colors, there exists an x-y hamiltonian path; or else (b) it is nearly equitable and whenever x and y are nodes of the larger color set, there exists an x-y hamiltonian path. Hsieh et al. [8] extended this concept into strongly hamiltonian laceability. A hamiltonian laceable graph G is strongly hamiltonian laceable if (a) G is equitable and there is a simple path of length |V (G)| − 2 between any two nodes of the same color; or else (b) G is nearly equitable and there is a simple path of length |V (G)| − 2 between any two nodes of opposite colors. Lewinter et al. [11] further introduced the concept of hyper-hamiltonian laceability. A hamiltonian laceable graph G is hyper-hamiltonian laceable if (a) G is equitable and if v is any node of G, G − v is hamiltonian laceable; or else (b) G is nearly equitable and if v is any node in the large color set, G − v is hamiltonian laceable. Hamiltonian laceability, which deals with embedding a hamiltonian path in a given 2.

(3) graph, is an important topic in interconnection networks. The ring embedding problem, which deals with all the possible lengths of cycles in a given graph, is investigated in the interconnection networks [4, 1, 6, 9, 5]. A graph is called pancyclic if it contains a cycle of every length from 3 to |V (G)| inclusive [3]. The concept of pancyclicity has been extended to bipancyclicity [13]. Bipancyclicity is essentially a restriction of the concept of pancyclicity to bipartite graphs whose cycles are necessarily of even length. A bipartite graph is edge-bipancyclic [13] if every edge lies on a cycle of every even length from 4 to |V (G)| inclusive. Recent studies have proposed several operations performing on hamiltonian laceable graphs to yield several attractive properties. In [7], Harary and Lewinter proposed some recursively defined hamiltonian laceable graphs, denoted by J(G), and asked that whether there are additional operations performing on hamiltonian laceable graphs to yield hamiltonian laceability. In 1993, Lewinter [11] proposed an operation performing on hamiltonian laceable graphs to yield hyper-hamiltonian laceability. In 1996, Liu [12] proposed another recursively construction scheme to construct hamiltonian-type graphs. In Section 2, we show that the construction scheme proposed in [7] for J(G) will make hamiltonian laceable graphs to be both hamiltonian laceability and strongly hamiltonian laceability. We also propose some recursively construction schemes to construct hamiltonian-type graphs. On the other hand, bipancyclic property is also an attractive topic in interconnection networks. In 1988, Saad and Schultz [14] proved that hypercube, Qn , is bipancyclic if and only if n ≥ 2. In 1991, Jwo et al. [10] also showed that all the even cycles with length l such that 6 ≤ l ≤ n! can be embedded in star graph Sn . Hence, in Section 3, we show that these recursively construction schemes performing on bipancyclic and edgebipancyclic graphs can yield bipancyclicity and edge-bipancyclicity, respectively. Section 4 provides some concluding remarks.. 3.

(4) 2. Hamiltonian Laceability, Strongly Hamiltonian Laceability, and Hyper-Hamiltonian Laceability. Given a hamiltonian laceable graph G, Harary and Lewinter [7] proposed a construction scheme to extend G to a larger graph J(G), while maintaining the hamiltonian laceable property as follows. Let G be a bipartite graph with white and black node sets {x1 , x2 , · · · , xm } and {y1 , y2 , · · · , yn }, respectively. Let J(G) be the graph obtained from G by adding a white node w, a black node b, and all the edges (b, xi ) and (w, yj ) for 1 ≤ i ≤ m and 1 ≤ j ≤ n. See Figure 1.. graph G. x1. x2. x3. xm. w. y1. y2. y3. yn. b. Figure 1: Graph J(G). In [7], Harary and Lewinter showed that if G is an equitable hamiltonian laceable graph with at least four nodes, then J(G) is hamiltonian laceable. In the following, we have a further result that J(G) is not only hamiltonian laceable, but also strongly hamiltonian laceable. Theorem 1 Let G be an equitable hamiltonian laceable graph with at least four nodes. Then J(G) is strongly hamiltonian laceable. Moreover, J(G) − b and J(G) − w are both hamiltonian laceable. Proof. By [7], J(G) is hamiltonian laceable. To show that J(G) is strongly hamiltonian laceable, it is sufficient to prove that both J(G)−b and J(G)−w are hamiltonian laceable. We shall prove the case for J(G) − b. As for J(G) − w, the case is similar. 4.

(5) Case 1: For any xk , find a hamiltonian path of J(G) − b joining w and xk . (See Figure 2(a).) Since G is hamiltonian laceable, G has a hamiltonian path joining xk and yi for any i. We arbitrarily choose a yi and let xk , P, yi be a hamiltonian path of G. By definition, (w, yi) ∈ E(J(G)). Thus, J(G) − b has a hamiltonian path xk , P, yi, w. Case 2: For any xk = xk , find a hamiltonian path of J(G) − b joining xk and xk . (See Figure 2(b).) Since G is hamiltonian laceable, for any yi , G has a hamiltonian path joining xk and yi . We arbitrarily choose a yi and let yr ,ys be the two nodes adjacent to xk on this hamiltonian path. Hence, we may label this path asxk , P1 , yr , xk , ys , P2 , yi without loss of generality. By definition, (w, yr ), (w, yi) ∈ E(J(G)). So xk , P1 , yr , w, yi, P2 , ys , xk is a hamiltonian path in J(G) − b joining xk and xk . Therefore, the proof of this theorem is 2. complete.. graph G. xk. graph G. xk. w P. xk’. w P2. P1 yi. yr ys. b. (a). yi. b. (b). Figure 2: (a) Case 1: For any xk , find a hamiltonian path of J(G) − b joining w and xk . (b) Case 2: For any xk = xk , find a hamiltonian path of J(G) − b joining xk and xk . With a similar argument as above, we have the following result. Theorem 2 Let G be a nearly equitable hamiltonian laceable graph with at least five nodes. Then, J(G) is strongly hamiltonian laceable. 5.

(6) Now, we define a new graph J (G) with V (J (G)) = V (J(G)) and E(J (G)) = E(J(G))∪{(w, b)}. With this additional edge (w, b) in J (G), we have another stronger result that J (G) is hyper-hamiltonian laceable if G is hamiltonian laceable with at least four nodes. We note, however, that J(G) is not necessarily hyper-hamiltonian laceable. For example, let G be the complete bipartite graph K2,2 , J(G) is strongly hamiltonian laceable but not hyper-hamiltonian laceable. In order to prove that J (G) is hyper-hamiltonian laceable, we show a lemma first. Lemma 1 Let G be a bipartite graph. Suppose that G − v is hamiltonian laceable for every v ∈ V (G) if G is equitable or else G − v is hamiltonian laceable for every node v in the larger color set if G is nearly equitable, then G is hamiltonian laceable. Proof. We shall prove that G is hamiltonian laceable by finding a hamiltonian path (1) joining any two distinct nodes x and y with different colors if G is equitable, or else (2) joining any two distinct nodes x and y in the larger color set if G is nearly equitable. Let a be a node adjacent to y. Since G − y is hamiltonian laceable, G − y has a hamiltonian path joining x and a. Thus, there is a hamiltonian path of G joining x and y since (a, y) ∈ E(G). So G is hamiltonian laceable.. 2. Theorem 3 If G is an equitable hamiltonian laceable graph with at least four nodes, then J (G) is hyper-hamiltonian laceable. Proof. We shall prove the following two statements: (1) J (G) is hamiltonian laceable; and (2) J (G) − f is hamiltonian laceable for any f ∈ V (J (G)). By Lemma 1, (1) is correct if (2) is. Thus, we need only to check whether (2) holds. By Theorem 1, J (G) − f is hamiltonian laceable, if f = w or f = b. Now, consider the case that f = w and f = b, we may without loss of generality assume that f = ym for some m. Then, J (G) − ym contains a subgraph isomorphic to J (G) − b, where node b replaces the node ym . J (G) − b is hamiltonian laceable, so is J (G) − ym . Thus, the 2. theorem follows. 6.

(7) In a like manner as above, we have the following result. Theorem 4 If G is a nearly equitable hamiltonian laceable graph with at least five nodes, then J (G) is hyper-hamiltonian laceable. Therefore, we may use the above two operations, J and J , to recursively construct infinitely many hamiltonian-type graphs. In the following, we have yet another construction scheme for X(G), to construct hamiltonian laceable graphs. With operation X, we can recursively construct hamiltonian laceable graphs by adding less edges to G than J. Definition 1 Let G be a hamiltonian laceable graph with white node set {x1 , x2 , · · · , xm } and black node set {y1 , y2, · · · , yn }.. Let X(G) be the graph resulting from adding a. white node w and a black node b. And E(X(G)) = E(G) ∪ (w, b) ∪ (w, yn ) ∪ (b, xm )∪ . (xm ,yi )∈E(G). (w, yi) ∪. . (b, xi ). In other words, we arbitrarily choose a white node. (yn ,xi )∈E(G). xm and a black node yn , make a new copy for each, say w and b, and add necessarily edges together with three more edges (w, b), (w, yn ), and (b, xm ). See Figure 3.. x1. x2. x3. xm-1 xm. w. y1. y2. y3. yn-1 yn. b. Figure 3: Graph X(G). By the definition of X(G), we have the following result. Theorem 5 If G is an equitable hamiltonian laceable graph with at least four nodes, then X(G) is also hamiltonian laceable.. 7.

(8) Proof. Let V (G) be the union of white node set {x1 , x2 , · · · , xn } and black node set {y1 , y2 , · · · , yn }. To prove that X(G) is hamiltonian laceable, for any two nodes with different colors, we need to find a hamiltonian path joining these two. We divide the proof into three cases. Case 1: Find a hamiltonian path of X(G) joining w and b. (See Figure 4(a).) Since G is hamiltonian laceable, G has a hamiltonian path xn , P1 , yn . (b, xn ), (w, yn) ∈ E(X(G)) by definition. Thus, we have a hamiltonian path b, xn , P1 , yn , w of X(G). Case 2: For any yi and xi , find a hamiltonian path of X(G) joining w and yi , and a hamiltonian path joining b and xi . (See Figure 4(b).) We shall show the case for w and yi , the other case is similar. Since G is hamiltonian laceable, we have a hamiltonian path of G joining xn and yi , say xn , P2 , yi. By definition, (b, xn ), (w, b) ∈ E(X(G)). Hence, w, b, xn , P2 , yi forms a hamiltonian path of X(G). Case 3: For any xi and yj , find a hamiltonian path of X(G) joining xi and yj . (See Figure 4(c).) Since G is hamiltonian laceable, we have a hamiltonian path of G joining xi and yj , say xi , P3 , yn , P4 , yj . Let xk be the node adjacent to yn on path P3 . We remark that if yj = yn , then P4 is an empty path. Thus, we may relabel xi , P3 , yn , P4 , yj as xi , P3 , xk , yn , P4 , yj . By definition, edges (xk , b), (b, w), and (w, yn ) belong to E(X(G)). Therefore, xi , P3 , xk , b, w, yn , P4 , yj is a hamiltonian path of X(G). Therefore, the proof 2. of this theorem is complete. With a similar argument as above, we have the following result.. Theorem 6 If G is a nearly equitable hamiltonian laceable graph with at least five nodes, then X(G) is also hamiltonian laceable.. 8.

(9) graph G. graph G. xn. graph G. xn. w. w. xi xk. w P4. P2. P1. P’3 yn. b. (a). yi. b (b). yn. yj. b. (c). Figure 4: (a) Case 1: Find a hamiltonian path of X(G) joining w and b; (b) Case 2: For any yi and xi , find a hamiltonian path of X(G) joining w and yi , and a hamiltonian path joining b and xi ; (c) Case 3: For any xi and yj , find a hamiltonian path of X(G) joining xi and yj .. 3. Bipancyclicity and Edge-Bipancyclicity. Theorem 7 Let G be an equitable bipartite graph with at least four nodes. If G is bipancyclic, then J(G) is also bipancyclic. Proof. We show this result by finding cycles of every even length from 4 to |V (G)| + 2 in J(G). Since G is bipancyclic, there are cycles of every even length from 4 to |V (G)| in J(G). Hence, we need only to find a cycle of length |V (G)| + 2 in J(G). Let a1 , a2 , a3 , and a4 be four consecutive nodes in the hamiltonian cycle of G; the color of a1 ,a3 be white, and a2 ,a4 be black. Thus, we may label this cycle as a1 , a2 , a3 , a4 , P, a1 . Then, a1 , b, a3 , a2 , w, a4, P, a1 forms a hamiltonian cycle of J(G) with length |V (G)| + 2. (See Figure 5.) Consequently, this theorem is proved.. 2. Theorem 8 Let G be an equitable bipartite graph with at least four nodes. If G is edgebipancyclic, then J(G) is also edge-bipancyclic. Proof. Let e be any edge in J(G). To prove that J(G) is edge-bipancyclic, we need to find cycles containing edge e of every even length from 4 to |V (G)| + 2. Case 1: e ∈ E(G). 9.

(10) graph G. w. a4 P. a1. a3 a2 b. Figure 5: Pancyclicity of J(G). Let e = (xi , yj ) for some i, j. By the assumption that G is edge-bipancyclic, J(G) has cycles containing edge (xi , yj ) of every even length from 4 to |V (G)|. Let y, xi, yj , x be four consecutive nodes in the hamiltonian cycle of G containing edge (xi , yj ). By definition, edges (w, y), (w, yj ), (b, x), and (b, xi ) belong to E(J(G)). Then, y, w, yj , xi , b, x, P1 , y forms a hamiltonian cycle of J(G) with length |V (G)| + 2. (See Figure 6(a).) Case 2: e ∈ E(J(G)) − E(G). Without loss of generality, we may consider only the case that e = (b, xi ) for any i. Consequently, we will find a cycle containing edge (b, xi ) of every even length from 4 to |V (G)| + 2 in J(G). Since G is bipancyclic, there are cycles of every even length from 4 to |V (G)| in G. Let xi , a2 , a3 , a4 be four consecutive nodes in this cycle; and the color of xi ,a3 be white, and a2 ,a4 be black. Thus, we may label this cycle as xi , a2 , a3 , a4 , P2 , xi . Therefore, xi , b, a3 , a2 , w, a4 , P2 , xi form cycles containing edge (b, xi ) of every even length from 6 to |V (G)|+2 of J(G). (See Figure 6(b).) In addition, xi , b, a3 , a2 , xi forms a cycle containing edge (b, xi ) of length 4 of J(G). (See Figure 6(c).) This theorem is complete. 2 In the following two theorems, we establish that X is also a recursively construction scheme for bipancyclic and edge-bipancyclic graphs. Theorem 9 Let G be an equitable bipartite graph with at least four nodes. If G is bipan10.

(11) graph G. graph G. graph G. w. y P1. x. w. a4. xi yj. P2. xi. a3 a2. b. a4 xi. w a3 a2 b. b. (a). (c). (b). Figure 6: Edge-bipancyclicity of J(G). cyclic, then X(G) is also bipancyclic. Proof. By the assumption that G is bipancyclic, to prove that X(G) is also bipancyclic, we need to find a cycle of length |V (G)| + 2 in X(G). Since G is equitable and bipancyclic, there will be a hamiltonian cycle in G. Let V (G) be the union of white nodes {x1 , x2 , · · · , xn } and black nodes {y1 , y2, · · · , yn }. Let xn and a be two consecutive nodes on this hamiltonian cycle. Thus, we may label this cycle as xn , a, P, xn . Therefore, xn , b, w, a, P, xn forms a hamiltonian cycle of length |V (G)| + 2 of X(G). (See Figure 2. 7.) This completes the proof.. graph G. w P. xn a b. Figure 7: Pancyclicity of X(G).. Theorem 10 Let G be an equitable bipartite graph with at least four nodes. If G is edge-bipancyclic, then X(G) is also edge-bipancyclic. 11.

(12) Proof. Let V (G) be the union of white node set {x1 , x2 , · · · , xn } and black node set {y1 , y2 , · · · , yn }. Let e be any edge in X(G). We shall prove that X(G) is edge-bipancyclic by finding cycles containing edge e of every even length from 4 to |V (G)| + 2 in X(G). Case 1: e ∈ E(G). Because G is edge-bipancyclic, we can find cycles of every even length from 4 to |V (G)| in G. Let yi, xn , yj , P1 , yi be a hamiltonian cycle of G containing edge e where yi , yj are two nodes adjacent to xn . Of course, at least one of (xn , yi ) and (yj , xn ) is not edge e, say (xn , yi) = e. By definition, edges (w, b), (w, yi), and (b, xn ) belong to E(X(G)). Thus, b, xn , yj , P1 , yi , w, b forms a cycle containing edge e of length |V (G)| + 2 in X(G). (See Figure 8(a).) Case 2: e = (w, b). Since G is edge-bipancyclic, we can find cycles of every even length from 4 to |V (G)| in G. We may label this cycle of G as xn , a, P2, xn where a is a neighbor of xn . Thus, xn , b, w, a, P2, xn is a cycle containing edge (w, b), so there are cycles containing edge (w, b) of every even length from 6 to |V (G)| + 2, and xn , b, w, a, xn is a cycle containing edge (w, b) of length 4. (See Figure 8(b).) Case 3: e = (w, yn ) or e = (b, xn ). The case holds for e = (b, xn ) as shown in Case 2. For e = (w, yn), it can be proved similarly. Case 4: e ∈. (xn ,yi )∈E(G). (w, yi) or e ∈. . (b, xi ).. (yn ,xi )∈E(G). Without loss of generality, we may consider only e = (w, yi) for some i such that (xn , yi) ∈ E(G). Of course, (xn , yi) ∈ E(G). Since G is edge-bipancyclic, there exist cycles containing edge (xn , yi) of every even length from 4 to |V (G)| in G. Let xn , yi, P3 , xn be one such cycle. Then, xn , b, w, yi, P3 , xn is a cycle containing edge (w, yi) with two 12.

(13) more edges. Thus, there are cycles containing edge (w, yi) of every even length from 6 to |V (G)| + 2 in X(G); and xn , b, w, yi, xn is a cycle containing edge (w, yi) of length 4. 2. (See Figure 8(c).) Therefore, the proof of this theorem is complete.. graph G. graph G. w. yj P1. graph G. xn. w xn. P2. yi. w P3. a b. (a). xn yi. b (b). b (c). Figure 8: Edge-bipancyclicity of X(G). With a similar argument, we have the following two results. Theorem 11 Let G be an equitable bipartite graph with at least four nodes. If G is bipancyclic, then J (G) is also bipancyclic. Theorem 12 Let G be an equitable bipartite graph with at least four nodes. If G is edge-bipancyclic, then J (G) is also edge-bipancyclic.. 4. Concluding Remarks. In this paper, we extend the result presented in [7] and we show that operation J performing on hamiltonian laceable graphs is not only hamiltonian laceable, but also strongly hamiltonian laceable. Furthermore, we show that by adding one more specific edge to J(G), the resulting graph J (G) becomes both strongly hamiltonian laceable and hyperhamiltonian laceable. We observe that J(G) and J (G) are two graphs by adding as many as O(|V (G)|) edges to G. Thus, we have another operation, X, adding considerably less number of edges to G to recursively construct hamiltonian laceable graphs.. 13.

(14) It is noticed that Hamiltonian laceability is an important topic which deals with embedding a hamiltonian path in a given graph. On the other hand, bipancyclicity and edge-bipancyclicity for bipartite graphs are important issues in interconnection networks. We show that the three operations, J, J , and X, performing on bipancyclic and edgebipancyclic graphs can yield bipancyclicity and edge-bipancyclicity, respectively.. References [1] V. Auletta, A. A. Rescigno, and V. Scarano, Embedding graphs onto supercubes, IEEE Trans. Comput 44 (4), 593–597, (1995). [2] J. A. Bondy and U. S. R. Murty, Graph theory with applications, North Holland, New York (1980). [3] J. A. Bondy, Pancyclic graphs I, Journal of Combinatorial Theory 11, 80–84, (1971). [4] K. Day and A. Tripathi, Embedding of cycles in arrangement graphs, IEEE Trans. Comput. 12, 1002–1006, (1993). [5] Jianxi Fan, Hamilton-connectivity and cycle-embedding of the Möbius cubes, Infor. Processing Letters 82, 113–117, (2002). [6] A. Germa, M. C. Heydemann, and D. Sotteau, Cycles in the cube-connected cycles graph, Discr. Appl. Math. 83, 135–155, (1998). [7] F. Harary and M. Lewinter, Hypercubes and other recursively defined Hamilton laceable graphs, Congressus Numerantium 60, 81–84, (1987). [8] S. Y. Hsieh, G. H. Chen, and C. W. Ho, Hamiltonian-laceability of star graphs, Networks 36, 225–232, (2000). [9] S. C. Hwang and G. H. Chen, Cycles in butterfly graphs, Networks 35 (2), 161–171, (2000).. 14.

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