On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

For a graph H , let σ t ( H ) = min { Σ i = 1 t d H ( v i ) | { v 1 , v 2 , … , v t } is an independent set in H } and let U t ( H ) = min { | ⋃ i = 1 t N H ( v i ) | | { v 1 , v 2 , ⋯ , v t } is an independent set in H } . We show that for a given number ϵ and given integers p ≥ t \u3e 0 , k ∈ { 2 , 3 } and N = N ( p , ϵ ) , if H is a k -connected claw-free graph of order n \u3e N with δ ( H ) ≥ 3 and its Ryjác̆ek’s closure c l ( H ) = L ( G ) , and if d t ( H ) ≥ t ( n + ϵ ) ∕ p where d t ( H ) ∈ { σ t ( H ) , U t ( H ) } , then either H is Hamiltonian or G , the preimage of L ( G ) , can be contracted to a k -edge-connected K 3 -free graph of order at most max { 4 p − 5 , 2 p + 1 } and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large: (i) If k = 2 , δ ( H ) ≥ 3 , and for a given t ( 1 ≤ t ≤ 4 ), then either H is Hamiltonian or c l ( H ) = L ( G ) where G is a graph obtained from K 2 , 3 by replacing each of the degree 2 vertices by a K 1 , s ( s ≥ 1 ). When t = 4 and d t ( H ) = σ 4 ( H ) , this proves a conjecture in Frydrych (2001). (ii) If k = 3 , δ ( H ) ≥ 24 , and for a given t ( 1 ≤ t ≤ 10 ) d t ( H ) \u3e t ( n + 5 ) 10 , then H is Hamiltonian. These bounds on d t ( H ) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved σ t and U t for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max { 4 p − 5 , 2 p + 1 } are fixed for given p , improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer

A Survey of Line Graphs and Hamiltonian Paths

In this paper, we are going to explore a survey of line graphs and hamiltonian paths. Research concerning line graphs and hamiltonian paths started in the 1960\u27s. We will investigate some recent theorems and proofs covering this topic. At the end, we will prove a main result involving line graphs and hamiltonian paths

Degree Conditions for Hamiltonian Properties of Claw-free Graphs

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 well-defined 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 claw-free 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 2-connected claw-free 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 2-connected claw-free 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

How tough is toughness?

The concept of toughness was introduced by Chvátal [34] more than forty years ago. Toughness resembles vertex connectivity, but is different in the sense that it takes into account what the effect of deleting a vertex cut is on the number of resulting components. As we will see, this difference has major consequences in terms of computational complexity and on the implications with respect to cycle structure, in particular the existence of Hamilton cycles and k-factors