Related papers: Further results on the Hamilton-Waterloo problem
In this paper, we construct almost resolvable cycle systems of order $4k+1$ for odd $k\ge 11$. This completes the proof of the existence of almost resolvable cycle systems with odd cycle length.
In this paper we give a complete solution to the Hamilton-Waterloo problem for the case of Hamilton cycles and C4k-factors for all positive integers k.
In this paper we consider the Hamilton cycle problem in the rectangular meshes with at most two faulty nodes.We prove that this problem is solvable in polynomial time with a corresponding algorithm. We provided an entirely new approach to…
The Hamilton-Waterloo Problem HWP$(v;m,n;\alpha,\beta)$ asks for a 2-factorization of the complete graph $K_v$ or $K_v-I$, the complete graph with the edges of a 1-factor removed, into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors, where…
It is conjectured that for every pair $(\ell,m)$ of odd integers greater than 2 with $m \equiv 1\; \pmod{\ell}$, there exists a cyclic two-factorization of $K_{\ell m}$ having exactly $(m-1)/2$ factors of type $\ell^m$ and all the others of…
We give a sharply-vertex-transitive solution of each of the nine Hamilton-Waterloo problems left open by Danziger, Quattrocchi and Stevens.
The Hamilton-Waterloo problem with uniform cycle sizes asks for a $2-$ factorization of the complete graph $K_v$ (for odd {\em v}) or $K_v$ minus a $1-$factor (for even {\em v}) where $r$ of the factors consist of $n-$cycles and $s$ of the…
As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion…
In this paper, we almost completely solve the Hamilton-Waterloo problem with C8- factors and Cm-factors where the number of vertices is a multiple of 8m.
We proposed an algorithm that covers some cases of Hamilton Circuit Problem.
We develop novel methods for constructing nearly Hamilton cycles in sublinear expanders with good regularity properties, as well as new techniques for finding such expanders in general graphs. These methods are of independent interest due…
We prove the existence of normally hyperbolic invariant cylinders in nearly integrable hamiltonian systems.
An instance of Hamiltonian cycle problem can be solved by converting it to an instance of Travelling salesman problem, assigning any choice of weights to edges of the underlying graph. In this note we demonstrate that, for difficult…
This paper concerns the existence of multiple rotating quasi-periodic solutions for second order Hamiltonian systems with sub-quadratic potential. Such solutions have the form $x(t+T)=Qx(t)$ for some orthogonal matrix $Q$. To deal with such…
We study the Hamilton cycle problem with input a random graph G=G(n,p) in two settings. In the first one, G is given to us in the form of randomly ordered adjacency lists while in the second one we are given the adjacency matrix of G. In…
We show the existence of homoclinic type solutions of second order Hamiltonian systems with a potential satisfying a relaxed superquadratic growth condition and a forcing term that is sufficiently small in the space of square integrable…
A. Gasull shared a list of 33 open problems in low dimensional dynamical systems in his work in 2021. The second part of Problem 3 is about whether the limit cycle of a quasi-homogeneous system $ \dot{x}=y,\; \dot{y}=-x^3+\alpha x^2y+y^3 $…
Some years ago I demonstrated a simulated annealing heuristic for the Hamiltonian cycle problem (Science 273, 413 (1996)). Here I propose an improved version of this heuristic.
A new class of accelerating, exact and explicit solutions of relativistic hydrodynamics is found - more than 50 years after the previous similar result, the Landau-Khalatnikov solution. Surprisingly, the new solutions have a simple form,…
In this paper, we will consider a kind of infinite dimensional Hamiltonian system(HS), by the method of saddle point reduction, topology degree and the index, we will get the existence of periodic solution for (HS).