For a vertex u of a graph G and an integer r, the ball of radius r centered at u is the subgraph Gr(u) induced by the set of all vertices of G whose distance from u does not exceed r. We investigate the set 1i of connected graphs G with at least 3 vertices such that every ball of radius 1 in G has a Hamilton cycle. We prove that every graph G in 1-£ with n vertices has at least 2n 3 edges, and ...

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdős-Rényi random graph Gn,p is around p ∼ logn+log logn n . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3-uniform...

A Hamilton cycle in a graph Γ is a cycle passing through every vertex of Γ. A Hamiltonian decomposition of Γ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph Kn on an odd number of vertices n has a Hamiltonian decomposition. This result was recently greatly extended b...

Since first introduced by Sudakov and Vu in 2008, the study of resilience problems random graphs received a lot attention probabilistic combinatorics. Of particular interest are spanning structures. It is known that for structures which contain many triangles, local cannot prevent an adversary from destroying all copies structure removing negligible amount edges incident to every vertex. In thi...

The Quadratic Travelling Salesman Problem (QTSP) is to find a least cost Hamilton cycle in an edge-weighted graph, where costs are defined on all pairs of edges contained in the Hamilton cycle. This is a more general version than the commonly studied QTSP which only considers pairs of adjacent edges. We define a restricted version of QTSP, the k-neighbour TSP (TSP(k)), and give a linear time al...

A well-known conjecture of Grünbaum and Nash-Williams proposes that 4-connected toroidal graphs are hamiltonian. The corresponding results for 4-connected planar and projective-planar graphs were proved by Tutte and by Thomas and Yu, respectively, using induction arguments that proved a stronger result, that every edge is on a hamilton cycle. However, this stronger property does not hold for 4c...

A famous conjecture of Pósa from 1962 asserts that every graph on n vertices and with minimum degree at least 2n/3 contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Komlós, Sárközy and Szemerédi [17]. We prove a degree sequence version of Pósa’s conjecture: Given any η > 0, every graph G of sufficiently large order n contains the square of a Hamilton...

The Hamilton cycle problem is closely related to a series of famous problems and puzzles (traveling salesman problem, Icosian game) and, due to the fact that it is NP-complete, it was extensively studied with different algorithms to solve it. The most efficient algorithm is not known. In this paper, a necessary condition for an arbitrary un-directed graph to have Hamilton cycle is proposed. Bas...