Let G = (V (G), E(G)) be a finite simple graph without loops. The neighbourhood N (v) of a vertex v is the set of vertices adjacent to v. The degree d(v) of v is |N (v)|. The minimum and maximum degree of G are denoted by δ(G) and 1(G), respectively. For a vertex v ∈ V (G) and a subset S ⊆ V (G), NS(v) is the set of neighbours of v contained in S, i.e., NS(v) = N (v) ∩ S. We let dS(v) = |NS(v)|...

This research develops a polynomial time algorithm for Hamilton Cycle(Path) and proves its correctness. A program is developed according to this algorithm and it works very well. This paper declares the research process, algorithm as well as its proof, and the experiment data. Even only the experiment data is a breakthrough.

An edge-colored graph H is properly colored if no two adjacent edges of H have the same color. In 1997, J. Bang-Jensen and G. Gutin conjectured that an edgecolored complete graph G has a properly colored Hamilton path if and only if G has a spanning subgraph consisting of a properly colored path C0 and a (possibly empty) collection of properly colored cycles C1, C2, . . . , Cd such that V (Ci) ...

The nonexponential Schilder-type theorem in Backhoff-Veraguas, Lacker, and Tangpi [Ann.\Appl. Probab., 30 (2020), pp. 1321--1367] is expressed as a convergence result for path-dependent partial differential equations with appropriate notions of generalized solutions. This entails non-Markovian counterpart to the vanishing viscosity method. We show uniqueness maximal subsolutions viscous Hamilto...

We consider the derangements graph in which the vertices are permutations of f1 : : : ng. Two vertices are joined by an edge if the corresponding permutations diier in every position. The derangements graph is known to be hamil-tonian and it follows from a recent result of Jung that every pair of vertices is joined by a Hamilton path. We use this result to settle an open question, by showing th...

We study M-alternating Hamilton paths, and M-alternating Hamilton cycles in a simple connected graph G on ν vertices with a perfect matchingM . Let G be a bipartite graph, we prove that if for any two vertices x and y in different parts of G, d(x)+d(y) ≥ ν/2+2, then G has an M-alternating Hamilton cycle. For general graphs, a condition for the existence of an M-alternating Hamilton path startin...

In this paper, we build on the work of Alspach, Chen, and Dean [2] who showed that proving the hamiltonicity of the Cayley graph of the the dihedral group Dn reduces to showing that certain cubic, connected, bipartite graphs (called honeycomb toroidal graphs) are hamilton laceable; that is, any two vertices at odd distance from each other can be joined by a hamilton path. Alspach, Chen, and Dea...

In this article current directions in solving Lovász’s problem about the existence of Hamilton cycles and paths in connected vertex-transitive graphs are given. © 2009 Elsevier B.V. All rights reserved. 1. Historical motivation In 1969, Lovász [59] asked whether every finite connected vertex-transitive graph has a Hamilton path, that is, a simple path going through all vertices, thus tying toge...

We consider the derangements graph in which the vertices are permutations of f ng Two vertices are joined by an edge if the corresponding permuta tions di er in every position The derangements graph is known to be hamil tonian and it follows from a recent result of Jung that every pair of vertices is joined by a Hamilton path We use this result to settle an open question by showing that it is p...