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On a hamiltonian cycle in which specified vertices are not isolated
Abstract
Let G be a graph with n vertices and minimum degree at least n/2, and B a set of vertices with at least 3n/4 vertices. In this paper, we show that there exists a hamiltonian cycle in which every vertex in B is adjacent to some vertex in B.
... Hamilton graph's judgment is very difficult, it has proven to be a NP (nondeterministic polynomial) problem. Currently, there are a lot of relevant researches [3][4][5][6][7][8][9][10][11][12][13] . Through ingenious application of Hamilton graph's characteristics, the Paper presents an efficient heuristic determination algorithm, which helps make a quick judgment for the specified undirected graph. ...
Determinating whether a graph is Hamiltonian is a open difficult problem. In this paper, the problem is converted to determinating whether the graph has a 2-regular Hamiltonian spanning subgraph. We also give the procedures of the method, which can be used directly on computer.
Dirac and Ore-type degree conditions are given for a graph to contain vertex disjoint cycles each of which contains a previously specified edge. One set of conditions are given that imply vertex disjoint cycles of length at most 4, and another set of conditions are given that imply the existence of cycles that span all of the vertices of the graph (i.e. a 2-factor). The conditions are shown to be sharp and give positive answers to conjectures of H. Enomoto in [private communication (1997)] and H. Wang [J. Graph Theory 26, No. 2, 105-109 (1997; Zbl 0886.05093)].
Let G be a graph with n vertices and minimum degree at least n/2, and let d be a positive integer such that d⩽n/4. We define a distance between two vertices as the number of edges of a shortest path joining them. In this paper, we show that, for any vertex subset A with at most n/2d vertices, there exists a Hamiltonian cycle in which the distance between any two vertices of A is at least d.
. Let G be a connected n--order graph and suppose that n = n 1 + Delta Delta Delta + n k where n i 2 are integers. The following will be proved : If G has minimum degree at least Xi 1 2 n 1 Pi + Delta Delta Delta + Xi 1 2 n k Pi then G has a spanning subgraph which consists of paths of orders n 1 ; : : : ; n k . 1. Introduction All graphs considered in this paper will be simple. V (G) and E(G) will denote the set of vertices and edges respectively in a graph G. A subgraph of G is said to be spanning if it contains all vertices of G. A path P with V (P ) = fx 1 ; x 2 ; : : : ; xm g and E(P ) = fx i x i+1 : 1 i m Gamma 1g will also be written as P = (x 1 ; x 2 ; : : : ; xm ). If P is ordered we denote the successor (predecessor) of a vertex x by x + (x Gamma ), and inductivly x +i the successor of x +(iGamma1) . The neighbourhood N(v) of a vertex v is the set of vertices adjacent to v, and the degree d(v) of v is jN(v)j, the size of the neighbour...