A sufficient condition for pancyclability of graphs

AbstractLet G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by dS(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with dS(u,v)=2, max{d(u),d(v)}≥n2, then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with dS(u,v)=2, max{d(u),d(v)}≥n+12, then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221–227] for the case when S=V(G)

A look at cycles containing specified elements of a graph

AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration

Cyclability in bipartite graphs

Let $G=(X,Y,E)$ be a balanced $2$-connected bipartite graph and $S \subset V(G)$. We will say that $S$ is cyclable in $G$ if all vertices of $S$ belong to a common cycle in $G$. We give sufficient degree conditions in a balanced bipartite graph $G$ and a subset $S \subset V(G)$ for the cyclability of the set $S$

Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs

The study of cycles, particularly Hamiltonian cycles, is very important in many applications. Bondy posited his famous metaconjecture, that every condition sufficient for Hamiltonicity actually guarantees a graph is pancyclic. Pancyclicity is a stronger structural property than Hamiltonicity. An even stronger structural property is for a graph to be cycle extendable. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable. In this dissertation, cycle extendability is investigated and generalized. It is proved that chordal 2-connected K1,3-free graphs are cycle extendable. S-cycle extendability was defined by Beasley and Brown, where S is any set of positive integers. A conjecture is presented that Hamiltonian chordal graphs are {1, 2}-cycle extendable. Dirac’s Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Ore’s condition is another early result giving a sufficient condition for Hamiltonicity. In this dissertation, generalizations of Dirac’s and Ore’s Theorems are presented. The Chvatal-Erdos condition is a result showing that if the maximum size of an independent set in a graph G is less than or equal to the minimum number of vertices whose deletion increases the number of components of G, then G is Hamiltonian. It is proved here that the Chvatal-Erdos condition guarantees that a graph is cycle extendable. It is also shown that a graph having a Hamiltonian elimination ordering is cycle extendable. The existence of Hamiltonian cycles which avoid sets of edges of a certain size and certain subgraphs is a new topic recently investigated by Harlan, et al., which clearly has applications to scheduling and communication networks among other things. The theory is extended here to bipartite graphs. Specifically, the conditions for the existence of a Hamiltonian cycle that avoids edges, or some subgraph of a certain size, are determined for the bipartite case. Briefly, this dissertation contributes to the state of the art of Hamiltonian cycles, cycle extendability and edge and graph avoiding Hamiltonian cycles, which is an important area of graph theory

A new sufficient condition for pancyclability of graphs

International audienceLet $G$ be a graph of order $n$ and $S$ be a set of $s$ vertices. We call $G$, $S$-pancyclable, if for every $i$ with $3\leq i \leq s$, there exists a cycle $C$ in $G$ such that $|V(C)\cap S|=i$. For any two nonadjacent vertices $u,v$ of $S$, we say that $u,v$ are of distance 2 in $S$, denoted by $d_S(u,v)=2$, if there is a path $P$ in $G$ connecting $u$ and $v$ such that $|V(P)\cap S|\leq 3$. In this paper, we will prove that: Let $G$ be a 2-connected graph of order $n$ and $S$ be a subset of $V(G)$ with $|S|\geq 3$. If $\max \{d(u),d(v)\} \geq n/2$ for all pairs of vertices $u,v$ of $S$ with $d_S(u,v)=2$, then $G$ is $S$-pancyclable or else $|S|=4r$ and $G[S]$ is a spanning subgraph of $F_{4r}$, or else $|S|=n$ is even and $G$ is the complete bipartite graph $K_{n/2,n/2}$, or else $|S|=n≥6$ is even and $G$ is $K'_{n/2,n/2}$, or else $G[S] = K_{2,2}$ and the structure of $G$ is well characterized. This generalizes a result of Benhocine and Wojda for the case when $S = V (G)$. [A. Benhocine, A.P. Wojda, The Geng-Hua Fan conditions for pancyclic or Hamilton-connected graph, J. Combin. Theory B 42 (1987) 167–180]