Degree-Light-Free Graphs and Hamiltonian Cycles

Theoretical Computer Science, 2010

Since finding whether a graph has a Hamiltonian path or Hamiltonian cycle are both NPcomplete problems, researchers have been formulating sufficient conditions that ensure the path or cycle. Rahman and Kaykobad [2] presented a sufficient condition for determining the existence of Hamiltonian path. Three recent works -Lenin Mehedy, Md. Kamrul Hasan, Mohammad Kaykobad (2007) [3], Rao Li (2006) [4], Shengjia Li, Ruijuan Li, Jinfeng Feng [5] -further used the same or similar condition to ensure Hamiltonian cycle with some exceptions. The three works, along with their unique findings, have some common results. This paper unifies the results and brings them under Rahman and Kaykobad's condition.

Discussiones Mathematicae Graph Theory, 2004

In [2], Brousek characterizes all triples of graphs, G 1 , G 2 , G 3 , with G i = K 1,3 for some i = 1, 2, or 3, such that all G 1 G 2 G 3-free graphs contain a hamiltonian cycle. In [6], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G 1 , G 2 , G 3 , none of which is a K 1,s , s ≥ 3 such that G 1 , G 2 , G 3-free graphs of sufficiently large order contain a hamiltonian cycle. In this paper, a characterization will be given of all triples G 1 , G 2 , G 3 with none being K 1,3 , such that all G 1 G 2 G 3-free graphs are hamiltonian. This result, together with the triples given by Brousek, completely characterize the forbidden triples G 1 , G 2 , G 3 such that all G 1 G 2 G 3-free graphs are hamiltonian.

Information Processing Letters, 2007

A Hamiltonian cycle is a closed path through all the vertices of a graph. Since discovering whether a graph has a Hamiltonian path or a Hamiltonian cycle are both NP-complete problems, researchers concentrated on formulating sufficient conditions that ensure Hamiltonicity of a graph. A recent paper [Rahman M. Sohel and Kaykobad M., "On Hamiltonian Cycles and Hamiltonian Paths", Information Processing Letters 94 , 37-51] presents distance based sufficient conditions for the existence of a Hamiltonian path. In this paper we establish that the same condition forces Hamiltonian cycle to be present excepting for the case where end points of a Hamiltonian path is at a distance greater than 2.

Discrete Applied Mathematics, 1997

We prove that if a graph G on n > 32 vertices is hamiltonian and has two nonadjacent vertices u and u with d(u) + d(u) 3 n + z where z = 0 if n is odd and z = 1 if n is even, then G contains all cycles of length m where 3 < m < 1/5(n + 13).

Discrete Mathematics, 2002

A graph G is called triangle-free if G has no induced K3 as a subgraph. We set 3 = min{ 3 i=1 d(vi)|{v1; v2; v3} is an independent set of vertices in G}. In this paper, we show that if G is a 1-tough and triangle-free graph of order n with n 6 3, then G is hamiltonian.

Journal of Combinatorial Theory, Series B, 1977

Information Processing Letters, 2006

Rahman and Kaykobad proved the following theorem on Hamiltonian paths in graphs. Let G be a connected graph with n vertices. If d(u) + d(v) + δ(u, v) n + 1 for each pair of distinct non-adjacent vertices u and v in G, where δ(u, v) is the length of a shortest path between u and v in G, then G has a Hamiltonian path. It is shown that except for two families of graphs a graph is Hamiltonian if it satisfies the condition in Rahman and Kaykobad's theorem. The result obtained in this note is also an answer for a question posed by Rahman and Kaykobad.

Discrete Mathematics, 2001

A graph G is said to be n-factor-critical if G − S has a 1-factor for any S ⊂ V (G) with |S| = n. In this paper, we prove that if G is a 2-connected n-factor-critical graph of order p with 3(G)¿ 3 2 (p − n − 1), then G is hamiltonian with some exceptions. To extend this theorem, we deÿne a (k; n)-factor-critical graph to be a graph G such that G − S has a k-factor for any S ⊂ V (G) with |S| = n. We conjecture that if G is a 2-connected (k; n)-factor-critical graph of order p with 3(G)¿ 3 2 (p − n − k), then G is hamiltonian with some exceptions. In this paper, we characterize all such graphs that satisfy the assumption, but are not 1-tough. Using this, we verify the conjecture for k62.

2014

Let D be a strongly connected directed graph of order n ≥ 4 which satisfies the following condition (*): for every pair of non-adjacent vertices x, y with a common in-neighbour d(x) + d(y) ≥ 2n − 1 and min{d(x), d(y)} ≥ n − 1. In [2] (J. of Graph Theory 22 (2) (1996) 181-187)) J. Bang-Jensen, G. Gutin and H. Li proved that D is Hamiltonian. In [9] it was shown that if D satisfies the condition (*) and the minimum semi-degree of D at least two, then either D contains a pre-Hamiltonian cycle (i.e., a cycle of length n − 1) or n is even and D is isomorphic to the complete bipartite digraph (or to the complete bipartite digraph minus one arc) with partite sets of cardinalities of n/2 and n/2. In this paper we show that if the minimum out-degree of D at least two and the minimum in-degree of D at least three, then D contains also a Hamiltonian bypass, (i.e., a subdigraph is obtained from a Hamiltonian cycle by reversing exactly one arc).

Australas. J Comb., 2007

Let G be a graph and H be a subgraph of G. If G contains a hamiltonian cycle C such that E(C) ∩ E(H) is empty, we say that C is an H-avoiding hamiltonian cycle. Let F be any graph. If G contains an H-avoiding hamiltonian cycle for every subgraph H of G such that H ∼ = F , then we say that G is F -avoiding hamiltonian. In this paper, we give minimum degree and degree-sum conditions which ensure that a graph G is Favoiding hamiltonian for various choices of F . In particular, we consider the cases where F is a union of k edge-disjoint hamiltonian cycles or a union of k edge-disjoint perfect matchings. If G is F -avoiding hamiltonian for any such F , then it is possible to extend families of these types in G. Finally, we undertake a discussion of F -avoiding pancyclic graphs.