Longest cycles and restgraph in maximalnonhamitonian graphs

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2008, Vol. 53, N ◦ . 5, pp. 49-53 LONGEST CYCLES AND RESTGRAPH IN MAXIMALNONHAMITONIAN GRAPHS Vu Dinh Hoa Hanoi National University of Education Abstract. A graph is called a maximalnonhamiltonian graph if it is non- hamiltonian and will be hamiltonian by adding any new edge. This paper presents a conjecture that if C is a longest cycle in a maximalnonhamilto- nian graph G then G−C is a complete graph and shows that the conjecture is true for some classes of graphs. 1. Terminology and notation We consider only undirected graphs without loop or multiple edges. Our ter- minology is standard except as indicated. We begin by introducing some definitions and notation. Let ω(G) denote the number of components of a graph G. Following Chv¡tal [3] a graph G is 1-tough if |S| ≥ ω(G − S) for any nonempty subset S of the vertex set V (G) of G. G is called nontough if there exists a nonempty subset S such that |S| < ω(G − S). We shall denote the number of vertices of G by n. We often study special spanning graphs, for example the cubic graphs [8] or the hamiltonian cycles. A graph G is called a hamiltonian graph if G contains a hamil- tonian cycle, i.e.a cycle of length n. Otherwise, G is nonhamiltonian. Similarly, a path is called a hamiltonian path if it contains all vertices in G. A graph G is called a maximalnonhamiltonian graph if G is nonhamiltonian and will be hamiltonian by adding any new edge. Clearly, for every nonadjacent vertices u and v in a maximal- nonhamiltonian graph G there is a hamiltonian path joining u with v. A cycle C of G is a dominating cycle if G− C is an edgeless graph and G is called dominated by C or dominable. The length `(C) of a longest cycle C in a graph G, called the circumference of G, will be denoted by c(G). For k ≤ α the independent number we denote by σk the minimum value of the degree sum of any k pairwise nonadjacent vertices. For k > α let σk =∞. Instead of σ1 we use the more common notation δ. Following Skupi²n [14], a graph G is called homogeneously traceable if for each vertex x of G there is a Hamiltonian path beginning at x. As Skupi²n defined in [12], pi(G) denotes the path partition number of G, i.e. pi(G) = −1 if G is hamiltonian connected; and pi(G) = 0 if G is hamiltonian but not hamiltonian-connected; and 49 Vu Dinh Hoa pi(G) is the smallest number such that there exists a system of pairwise disjoint paths P1, ... , Pt containing all vertices of G if G is not hamiltonian. This parameter has been investigated by Goodman and Hedetniemi [5] and Boesch et al. [2]. In [12,13] some relations between path partition number and hamiltonicity of graphs are proved which allow us to get some upper bounds of pi(G). Similarly, we can ask for an upper bound of pi(G) if G satisfies toughness conditions in the well-known conjecture of Chv¡tal on 2-tough graphs. In fact, it is a special case of the question posed by Skupi²n in the problem 1 in [12]. In this paper we also consider a special class of graphs with pi(G) ≤ 2 as follows. Let G be a graph with a Hamilton cycle and let S be a non-empty subset of V (G). Then we have pi(G−S) ≤ |S| and therefore pi(G−S) ≤ |S| for all non-empty sets S ⊆ V (G) is a necessary condition for hamiltonicity. This condition appeared already in Haggkvist [6] and Schiermeyer's definition of path-tough graphs [11]. In order to preclude trivialities we put that definition as follows. A graph G is called path-tough if G has at least three vertices and pi(G− S) ≤ |S| for all nonempty sets S ≤ V (G). Clearly, every homogeneously traceable graph is a path-tough graph. And, we have pi(G) ≤ 2 if G is a path-tough graph since for an arbitrary vertex v of G the graph G− {v} contains a hamiltonian path. Maximalnonhamiltonian tough graphs fall into 1-tough graphs with total ver- tices (vertices joining to all other vertices) and path-tough graphs without total vertices. In [15], conditions for the existence of hamiltonian cycles in 1-tough graphs with total vertices are proved. But, the study for hamiltonian cycles in path-tough graphs [6,11] seem very difficult. 2. Results and conjecture We begin with a well-known theorem of Nash-Williams [7]. Theorem 2.1. Let G be a 2-connected graph on n vertices with δ ≥ (n+2)/3. Then every longest cycle in G is a dominating cycle. Bigalke and Jung [1] improved Theorem 2.1 under the assumption that G is 1-tough. Theorem 2.2. Let G be a 1-tough graph on n vertices with δ ≥ n/3. Then every longest cycle in G is a dominating cycle. The class of graphs with δ ≥ n 3 was also studied by Enomoto et al. in [4]. Let C be a longest cycle in a nonhamiltonian graph G. In the proof techniques for the existence of a hamiltonian cycle we often ask for the rest graph G − C. Clearly, it is very useful if we know about G − C for any longest cycle C in a 50 Longest cycles and restgraph in maximalnonhamiltonian graphs maximalnonhamiltonian graph G. For some special cases we can describe the rest graph G− C. A such result is proved in [10]. Theorem 2.3. Let G be a 2-connected graph on n vertices with δ ≥ n/3 and C be a longest cycle in it. Then G− C is either a complete graph or an edgeless graph. It was also shown in [10] that if δ ≥ n/3 and G is 2-connected then G− C is only dependent on G, namely: The definition of the class < shows that Theorems 2.4 is a generalization of Theorem 2.1 and Theorem 2.2. The class < is the union of the classes of graphs <1, ... , <5 and it takes only a polynomial time to know whether a given graph G ∈ < or not. <1 is the class of graphs on 3r vertices and with minimal degree δ = r ≥ 3, which are isomorphic to a graph G, where 2Kr−1 + K¯2 ⊆ G ⊆ (2Kr−1 ∪Kr) +K2; <2 is the class of graphs on 15 vertices and with minimal degree δ = 5, which are isomorphic to a graph G, where 4K3 + K¯3 ⊆ G ⊆ 4K3 +K3; <3 is the class of graphs on 3r vertices and with minimal degree δ = r ≥ 3, which are isomorphic to a graph G, where (r − 1)K2 + K¯r−1 ⊆ G ⊆ ((r − 1)K2 ∪K3) +Kr−1; <4 is the class of graphs on 3r+2 vertices and with minimal degree δ = r+1, which are isomorphic to a graph G, where 3Kr + K¯2 ⊆ G ⊆ 3Kr +K2; <5 is the class of graphs on 3r+2 vertices and with minimal degree δ = r+1 ≥ 3, which are isomorphic to a graph G, where (r + 1)K2 + K¯r ⊆ G ⊆ (r + 1)K2 +Kr. Theorem 2.4. Let G be a 2-connected graph on n ≥ 3 vertices with δ ≥ n/3. Then the following properties are equivalent: (a) G is a dominable graph, (b) Every longest cycle in G is a dominating cycle, (c) G /∈ < for a class < of special graphs. 51 Vu Dinh Hoa Note that all graphs of < are nontough graphs, we can easily see that the graph G−C is a complete graph for any graph G ∈ < and every longest cycle C in G. Thus, we believe that: Conjeture 2.1. If G is a maximalnonhamiltonian graph and C is any longest cycle in G, then G− C is a complete graph. In the following, we show that the conjecture is true for nontough maximal- nonhamiltonian graphs. Theorem 2.5. Let C be a longest cycle in a nontough maximalnonhamiltonian graph G, then G− C is a complete graph. Proof. Since G is nontough, there exists a set S of vertices such that ω(G − S) ≥ |S| + 1. Let s = |S| and G1, G2, ... ,Gm be the components of G − S with m ≥ s + 1. Clearly, G1, G2, ... ,Gm are the complete graphs and m = s + 1 since G is a maximalnonhamiltonian graph. Otherwise, we can add a new edge in some Gi or a new edge joining two of the components of G− S to obtain a new nonhamiltonian graph with more edges. Similarly, S is a complete graph and every vertex of S is joining with any vertex of G−S. Now, we can easily see that for every longest cycle C the graph G−C is one of the components G1, G2, ... ,Gm, so complete graph. Now we will show that the same result holds for the class of graphs with δ ≥ n 3 . We denote by p(G) the length of a longest path in G. In [9], Van den Heuvel proved the following result. Lemma 2.1. (Corollary 6.8 in [9]) Let G be a 1-tough graph on n ≥ 3 vertices such that σ3 ≥ n. Then G satisfies c(G) ≥ p(G). Thus, we conclude the following result. Theorem 2.6. Let G be a maximalnonhamiltonian graph on n ≥ 3 vertices such that δ ≥ n 3 and C be a longest cycle in G, then G− C is a complete graph. Proof. If G is a nontough graph, then G− C is a complete one from Theorem 2.5. 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