121 research outputs found
A generalization of Ore's Theorem involving neighborhood unions
AbstractLet G be a graph of order n. Settling conjectures of Chen and Jackson, we prove the following generalization of Ore's Theorem: If G is 2-connected and |N(u)āŖN(v)|ā©¾12n for every pair of nonadjacent vertices u,v, then either G is hamiltonian, or G is the Petersen graph, or G belongs to one of three families of exceptional graphs of connectivity 2
Long cycles in graphs with large degree sums and neighborhood unions
We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings
Long cycles, degree sums and neighborhood unions
AbstractFor a graph G, define the parameters Ī±(G)=max{|S| |S is an independent set of vertices of G}, Ļk(G)=min{āki=1d(vi)|{v1,ā¦,vk} is an independent set} and NCk(G)= min{|āŖki=1 N(vi)ā„{v1,ā¦,vk} is an independent set} (kā©¾2). It is shown that every 1-tough graph G of order nā©¾3 with Ļ3(G)ā©¾n+rā©¾n has a cycle of length at least min{n,n+NCr+5+ā(n+r)(G)-Ī±(G)}, where Īµ(i)=3(ā13iāā13i). This result extends previous results in Bauer et al. (1989/90), FaĆbender (1992) and Flandrin et al. (1991). It is also shown that a 1-tough graph G of order nā©¾3 with Ļ3(G)ā©¾n+rā©¾n has a cycle of length at least min{n,2NCā18(n+6r+17)ā(G)}. Analogous results are established for 2-connected graphs
Long cycles in graphs containing a 2-factor with many odd components
We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+d(c)+d(w)n+2 for every tiple u, v, w of independent vertices. As a corollary we obtain the follwing improvement of a conjectre of HƤggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume G has a 2-factor with at least k+1 odd components. Then G is hamiltonian
A generalization of a result of HƤggkvist and Nicoghossian
Using a variation of the Bondy-ChvĆ”tal closure theorem the following result is proved: If G is a 2-connected graph with n vertices and connectivity Īŗ such that d(x) + d(y) + d(z) ā„ n + Īŗ for any triple of independent vertices x, y, z, then G is hamiltonian
Cycle spectra of Hamiltonian graphs
AbstractWe prove that every Hamiltonian graph with n vertices and m edges has cycles with more than pā12lnpā1 different lengths, where p=mān. For general m and n, there exist such graphs having at most 2āp+1ā different cycle lengths
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.
Diracās Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Oreās condition is another early result giving a sufficient condition for Hamiltonicity. In this dissertation, generalizations of Diracās and Oreās 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
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