View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 120 (2002) 219–237 Vertex pancyclic graphs Bert Randeratha; ∗,Ingo Schiermeyer b,Meike Tewes b,Lutz Volkmann c aInstitut fur Informatik, Universitat zu Koln, Pohligstrasse 1 50969 Koln, Germany bFakultat fur Mathematik und Informatik, TU Bergakademie Freiberg, 09596 Freiberg, Germany cLehrstuhl II fur Mathematik, RWTH Aachen, 52056 Aachen, Germany Received 17 September 1999; received in revised form 31 July 2000; accepted 21 July 2001 Abstract Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3 6 k 6 n,and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3 6 k 6 n. In this paper,we shall present di0erent su1cient conditions for graphs to be vertex pancyclic. ? 2002 Elsevier Science B.V. All rights reserved. 1. Terminology We consider ÿnite,undirected,and simple graphs G with the vertex set V (G) and the edge set E(G). The order of a graph G is denoted by n and the size by m, respectively. The complete graph of order n is denoted by Kn,and the complete bipartite graph with the partite sets A and B with |A| = p and |B| = q is denoted by Kp;q. A graph G is k-connected if G − S is connected for every subset S ⊆ V (G) with |S| 6 k − 1. A vertex v of a connected graph G is called a cut vertex if G − v is disconnected. The distance dG(u; v)=d(u; v) of two vertices u and v in G is the length of a shortest uv-path,which is a path connecting u and v. For A ⊆ V (G) let G[A]be the subgraph induced by A. The set of all neighbours of a vertex x ∈ V (G) is denoted by N(x)=N (x),and d(x)=d (x)=|N(x)| is the degree of the vertex x. We write G G N[x]=NG[x] instead of N(x)∪{x}. In addition,we deÿne N(X )=NG(X )= x∈X N(x) and N[X ]=NG[X ]=N(X )∪X for a subset X of V (G). We denote by (G) and (G) the minimum and maximum degree of a graph G,respectively. Let 2(G) denote the minimum degree sum of a pair of distinct nonadjacent vertices in G. A cycle Ck of length k is also called a k-cycle. A path of a graph G is a Hamiltonian path if it contains all the vertices of G. A graph G is said to be pancyclic if it has k-cycles for every k ∗ Corresponding author. E-mail address: [email protected] (B. Randerath). 0166-218X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S0166-218X(01)00292-X 220 B. Randerath et al. / Discrete Applied Mathematics 120 (2002) 219–237 between 3 and n. A vertex of a graph G is r-pancyclic if it is contained in a k-cycle for every k between r and n,and G is vertex r-pancyclic if every vertex is r-pancyclic. A 3-pancyclic vertex and a vertex 3-pancyclic graph are also called pancyclic and vertex pancyclic,respectively. Analogously, G is called edge r-pancyclic for some 3 6 r 6 n, if every edge of G belongs to a cycle of length k for every r 6 k 6 n. A graph G is Hamiltonian-connected,if for every pair u; v of distinct vertices of G, there exists a Hamiltonian path from u to v. It is called s-Hamiltonian,if G − S is Hamiltonian for every vertex set S with |S| 6 s. Following Hendry [13],we call a cycle C in a graph G extendable if there is a cycle C in G such that |V (C)| = |V (C)| + 1 and V (C) ⊆ V (C). If such a cycle C exists, we will say that C can be extended to C or that C is an extension of C. A graph G is cycle extendable if it has at least one cycle and every nonHamiltonian cycle in G is extendable. A graph G is fully cycle extendable if G is cycle extendable and every vertex of G lies on a cycle of length 3. In [6],Bondy and ChvÃatal introduced the closure of a graph and the stability of a property. The k-closure Clk (G) is obtained from G by recursively joining pairs of nonadjacent vertices whose degree sum is at least k,until no such pair remains. The k-closure is independent of the order of adjunction of the edges. Obviously,any graph of order n satisÿes G =Cl2n−3(G) ⊆ Cl2n−4(G) ⊆···⊆Cl1(G) ⊆ Cl0(G)=Kn: A property P deÿned on all graphs of order n is said to be k-stable if for any graph G of order n that does not satisfy P,the fact that uv is not an edge of G and that G + uv satisÿes P implies dG(u)+dG(v) ¡k. Vice versa,if uv ∈ E(G);dG(u)+dG(v) ¿ k and G + uv has property P,then G itself has property P. Every property is (2n − 3)-stable and every k-stable property is (k + 1)-stable. 2. Preliminaries and motivation In 1971 Bondy [4] posed the following metaconjecture: Almost any nontrivial condition on a graph which implies that the graph is Hamil- tonian also implies that it is pancyclic (except for maybe a simple family of exceptional graphs). For several su1cient conditions Bondys metaconjecture has been veriÿed. This is moti- vation to examine these su1cient conditions even for vertex pancyclicity,since vertex pancyclicity implies pancyclicity and pancyclicity implies Hamiltonicity. The ÿrst su1cient condition for a graph to be Hamiltonian is due to Dirac [9] in 1952. Theorem 1 (Dirac [9]). If G is a graph of order n ¿ 3 such that (G) ¿ n=2; then G is Hamiltonian. B. Randerath et al. / Discrete Applied Mathematics 120 (2002) 219–237 221 An interesting improvement of Diracs condition was obtained by Ore [14]. Theorem 2 (Ore [14]). Let G be a graph of order n ¿ 3. If 2(G) ¿ n; then G is Hamiltonian. The proof of Theorem 2 is based on the following more general result of Ore [14]. Lemma 3 (Ore [14]). Let u and v be distinct nonadjacent vertices of a graph G of order n ¿ 3 such that dG(u)+dG(v) ¿ n. Then G is Hamiltonian if and only if G+uv is Hamiltonian. Lemma 3 motivated Bondy and ChvÃatal [6] to the deÿnition of the closure. For recent overview on closure concepts the reader is referred to [8]. Diracs condition was also extended in the following direction. Theorem 4 (ErdJos [10]). Let G be a graph of size m and order n ¿ 3. If either = (G) ¿ n=2 or n − (n +2)=2 n − 1 2 m¿max + 2; + 2 2 2 then G is Hamiltonian. Theorem 5 (Ore [15]). Let G be a graph of order n ¿ 3. If 2(G) ¿ n +1; then G is Hamiltonian-connected. 3. Ore-type results We start with a lemma that we prove by using a classical argument (cf. for example Williamson [17] or Faudree et al. [11]). Lemma 6. Let G be a graph of order n ¿ 3 and 2(G) ¿ n +1. If the vertex w is not pancyclic; then dG(w) 6 (n − 1)=2. Proof. In view of Theorem 2, G is Hamiltonian,and G−w also contains a Hamiltonian cycle C(w). Let dG(w)=s and x1;x2;:::;xs be the neighbours of w.IfG contains no k-cycle C for some k with 3 6 k 6 n − 1 with w ∈ V (C),then no vertex yi,16 i 6 s, is adjacent to w,where yi is the vertex at distance k − 2 from xi on the arbitrarily oriented cycle C(w). Thus 2dG(w) 6 n − 1. Corollary 7 (Hendry [13]). Let G be a graph of order n ¿ 3 with (G) ¿ (n +1)=2. Then G is vertex pancyclic. Hendry has even shown in [13] that a graph G with (G) ¿ (n +1)=2 is fully cycle extendable. 222 B. Randerath et al. / Discrete Applied Mathematics 120 (2002) 219–237 We deÿne a very important family H;n of graph classes which will be useful in the following to provide bounds for vertex pancyclicity and fully extendability. Remark 8. A graph G of order n and with minimum degree is a member of the class H;n,if there exists a vertex w of minimum degree s.t. NG(w) induces an independent set and every vertex u ∈ NG(w) is adjacent to every vertex x ∈ V (G) − NG[w]. The unique member G ∈ H;n with the additional property that V (G) − NG[w] induces a c complete graph is denoted by H;n. The class Hn=2;n with n even shows that Corollary 7 is best possible,since every member G of Hn=2;n satisÿes (G) ¿ n=2 and it is easy to see that the vertex w of minimum degree is not contained in a 3-cycle. Theorem 9. Let G be a graph of order n ¿ 3 such that 2(G) ¿ 4n=3 −1. Then every vertex of G is contained in a 3-cycle.