Abstract
This thesis contains many new contributions to the field of hamiltonian graph theory, a very active subfield of graph theory. In particular, we have obtained new sufficient minimum degree and degree sum conditions to guarantee that the graphs satisfying these conditions, or their line graphs, admit a Hamilton cycle (or a Hamilton path), unless they have a small order or they belong to welldefined classes of exceptional graphs. Here, a Hamilton cycle corresponds to traversing the vertices and edges of the graph in such a way that all their vertices are visited exactly once, and we return to our starting vertex (similarly, a Hamilton path reflects a similar way of traversing the graph, but without the last restriction, so we might terminate at a different vertex).
In Chapter 1, we presented an introduction to the topics of this thesis together with Ryjáček’s closure for clawfree graphs, Catlin’s reduction method, and the reduction of the core of a graph. In Chapter 2, we found the best possible bounds for the minimum degree condition and the minimum degree sums condition of adjacent vertices for traceability of 2connected clawfree graph, respectively. In addition, we decreased these lower bounds with one family of well characterized exceptional graphs. In Chapter 3, we extended recent results about the conjecture of Benhocine et al. and results about the conjecture of Z.H Chen and H.J Lai. In Chapters 4, 5 and 6, we have successfully tried to unify and extend several existing results involving the degree and neighborhood conditions for the hamiltonicity and traceability of 2connected clawfree graphs.
Throughout this thesis, we have investigated the existence of Hamilton cycles and Hamilton paths under different types of degree and neighborhood conditions, including minimum degree conditions, minimum degree sum conditions on adjacent pairs of vertices, minimum degree sum conditions over all independent sets of t vertices of a graph, minimum cardinality conditions on the neighborhood union over all independent sets of t vertices of a graph, as well minimum cardinality conditions on the neighborhood union over all t vertex sets of a graph. Despite our new contributions, many problems and conjectures remain unsolved.
In Chapter 1, we presented an introduction to the topics of this thesis together with Ryjáček’s closure for clawfree graphs, Catlin’s reduction method, and the reduction of the core of a graph. In Chapter 2, we found the best possible bounds for the minimum degree condition and the minimum degree sums condition of adjacent vertices for traceability of 2connected clawfree graph, respectively. In addition, we decreased these lower bounds with one family of well characterized exceptional graphs. In Chapter 3, we extended recent results about the conjecture of Benhocine et al. and results about the conjecture of Z.H Chen and H.J Lai. In Chapters 4, 5 and 6, we have successfully tried to unify and extend several existing results involving the degree and neighborhood conditions for the hamiltonicity and traceability of 2connected clawfree graphs.
Throughout this thesis, we have investigated the existence of Hamilton cycles and Hamilton paths under different types of degree and neighborhood conditions, including minimum degree conditions, minimum degree sum conditions on adjacent pairs of vertices, minimum degree sum conditions over all independent sets of t vertices of a graph, minimum cardinality conditions on the neighborhood union over all independent sets of t vertices of a graph, as well minimum cardinality conditions on the neighborhood union over all t vertex sets of a graph. Despite our new contributions, many problems and conjectures remain unsolved.
Original language  English 

Awarding Institution  
Supervisors/Advisors 

Award date  5 Sept 2019 
Place of Publication  Enschede 
Publisher  
Print ISBNs  9789036546102 
DOIs  
Publication status  Published  5 Sept 2018 
Keywords
 Degree conditions
 Hamilton cycle
 Hamilton path
 Closure
 Reduction
 Clawfree graphs