arXiv subjects
Publications and source records attributed to Heping Jiang.
The Travelling Salesman Problem (TSP), finding a minimal weighted Hamilton cycle in a graph, is a typical problem in operation research and combinatorial optimization. In this paper, based on some novel properties on Hamilton graphs, we present a precise algorithm for finding a minimal weighted Hamilton cycle in a non-metric and symmetric graph with time complexity of \textit{O}(|E(G)|^3) , where |E(G)| is the size of graph G.
Deciding if a graph is a Hamilton graph, also named the Hamilton cycle problem, is important for discrete mathematics and computer science. Due to no characterization to identify Hamilton graphs effectively, there are no tractable algorithms to solve the Hamilton cycle problem. Grinberg Theorem is a necessary condition only for planar Hamilton graphs. In this paper, based on new studies on the Grinberg Theorem, in which we provided new properties of Hamilton graphs with respect to the cycle bases and improved the Grinberg Theorem to derive an efficient condition for Hamilton graphs, we present a new precise algorithm for deciding Hamilton graphs, named the '1+1' algorithm. Theoretically, the '1+1' algorithm terminates in $O(|E(G)|^3)$ worst time complexity, where $|\textit{E}(\textit{G})|$ is the size of the given graph $\textit{G}$.
Finding a Hamilton graph from simple connected graphs is an important problem in discrete mathematics and computer science. Grinberg Theorem is a well-known necessary condition for planar Hamilton graphs. It divides a plane into two parts: inside and outside faces. The sum of inside faces in a Hamilton graph is a Hamilton cycle. In this paper, using a basis of the cycle space to represent a graph and by the Inclusion-Exclusion Principle, we derive the equality with respect to the inside faces that can be also obtained from Grinberg Theorem. By further investigating the cycle structure of inside faces, we give a new combinatorial interpretation to Grinberg's condition, which explains why Grinberg Theorem is not sufficient for Hamilton graphs. Our results will improve deriving an efficient condition for Hamilton graphs.
A graph \textit{G} is a tuple (\textit{V}, \textit{E}), where \textit{V} is the vertex set, \textit{E} is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree 2 on an edge except for leaving only one vertex of degree 2. A 2-common (\textit{v}, \textit{0}) combination is a cycle set in which every pair of joint cycles \textit{A} and \textit{B} satisfies $|V(A)\cap V(B)|=2$ and $|E(A)\cap E(B)|=0$. In this paper, we investigate the cycle structure of 2-common (\textit{v}, \textit{0}) combination in reduced graphs, and give the characterizations of their Hamiltoncity.
A graph $\textit{G}$ is a tuple $(\textit{V}, \textit{E})$, where $\textit{V}$ is the vertex set, $\textit{E}$ is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree 2 on an edge except for leaving only one vertex of degree 2. We denote by I a set of cycles only jointed by inside vertices. |I| is the number of sets I in a graph. We use a norm graph to denote a reduced graph of |I|=1. $\textit{g}$ is a subgraph obtained by deleting all removable cycles from a basis of a norm graph. In this paper, we show that a norm graph $\textit{G}$ is non-Hamiltonian, if and only if, $\textit{g}$ and K_2_,_3 are homeomorphic.
Let G (V, E) be a simple graph with vertex set V and edge set E. A generalized cycle is a subgraph such that any vertex degree is even. A simple cycle (briefly in a cycle) is a connected subgraph such that every vertex has degree 2. A basis of the cycle space is called a cycle basis of G (V, E). A cycle basis where the sum of the weights of the cycles is minimal is called a minimum cycle basis of G. Grinberg theorem is a necessary condition to have a Hamilton cycle in planar graphs. In this paper, we use the cycles of a cycle basis to replace the faces and obtain an equality of inner faces in Grinberg theorem, called Grinberg equation. We explain why Grinberg theorem can only be a necessary condition of Hamilton graphs and apply the theorem, to be a necessary and sufficient condition, to simple graphs.
A Hamilton cycle is a cycle containing every vertex of a graph. A graph is called Hamiltonian if it contains a Hamilton cycle. The Hamilton cycle problem is to find the sufficient and necessary condition that a graph is Hamiltonian. In this paper, we give out some new kind of definitions of the subgraphs and determine the Hamiltoncity of edges according to the existence of the subgraphs in a graph, and then obtain a new property of Hamilton graphs as being a necessary and sufficient condition characterized in the connectivity of the subgraph that induced from the cycle structure of a given graph.
In this paper we extend general grid graphs to the grid graphs consist of polygons tiling on a plane, named polygonal grid graphs. With a cycle basis satisfied polygons tiling, we study the cyclic structure of Hamilton graphs. A Hamilton cycle can be expressed as a symmetric difference of a subset of cycles in the basis. From the combinatorial relations of vertices in the subset of cycles in the basis, we deduce the formula of inside faces in Grinberg theorem, called Grinberg equation, and derive a kind of cycles whose existence make a polygonal grid graph non-Hamiltonian, called non-Hamiltonian holes, and then we characterize the existence condition of non-Hamiltonian holes and obtain the necessary and sufficient condition of a polygonal grid graph to be Hamiltonian. The result in this paper provides a new starting point for developing a polynomial-time algorithm for Hamilton problem in general grid graphs.