Heavy Cyeles In Weighted Graphs And Ore Type Condition

All graphs considered in this paper are simple and undirected. Let G = (V(G),E(G)) be a graph, For v 6 V(G), d(v) denotes the number of vertices adjacent to a vertex v, F(u) denotes the set of vertices adjacent to a vertex v. G denotes the complement graph of a graph G; IG denotes the union graph of / copies of G. GI C G C GI means that G is isomorphic to a subgraph of G2 containing GI. For S V, 2, d(S) denotes the least number of edges of a path conneting two distinct vertices in S. When d(S) 2, S is the independent set of vertices in G.A graph G = (V,E) is called weighted graph, if each edge e is assigned a non-negative number to(e), called the weight of e. We define the weighted degree of v byIf G has a k-independent set of vertices in G, then defineindependent set of vertices in G}.otherwise, define The weighted of a subgraph H in G is defined bythe one with maximum weight among a certain kind of subgraphs is also called heaviest. In particular, a heaviest cycle is a cycle with maximum weight. Define the weighted circumference of a weighted graph G byv-cycle denotes a cycle which passes through a vertex v, e-cycle denotes a cycle which passes through an edge e.An unweighted graph can be regarded as a weighted graph in which each edge is assigned weight 1. Thus (circumference of a graph). A heaviest cycle is a longest cycle. Therefore, studying the weighted graph can obtain the more general result.Hamilton problem is one of fundamental problems in graph theory. If the number of vertices in a long cycle of a graph is just the number of vertices in the graph, the long cycle is exactly Hamiltonian cycle. So the research on long cycles is often involved in solving Hamilton problems. In this aspect, we have got many results. In 1952, Dirac gave the minimum degree condition on the existence of the longest cycle (Dirac type condition). In 1960, Ore gave the sum of degree condition(0re type condition). In some cases, Ore condition can get the better result. In 1984, Fan G.H. gave Fan type condition in [1].Research on the heaviest cycle in weighted graph is an important aspect. J. A.Bondy and Fan G.H. etc. studied the heaviest cycle in weighted graph and got many results in [2], [3]. In recent years, The following aspects are mainly focused on: Dirac type condition; Ore type condition; Fan type condition.In this paper, we mainly study heavy cycles in weighted graphs and Ore type condition.In the first section, we give a brief introduction on the basic concepts and terminology which are involved in this paper. All graphs considered in this paper are simple.In the second section, we generalize theorem 2.3 on the existence of the longest cycle of Ore type condition to weighted graphs and get theorem 2.5.Theorem 2.3 Suppose G is a 2-connected graph with, Then either G is Hamiltonian or through each vertex y of G there passes a cycle of length m.Theorem 2.5 Suppose G be 2 connected weighted graph with a? m , Then either G is Hamiltonian or through each vertex y of G there passes a cycle of weighted m.In order to make preparation for the proof of theorem 2.5, we first give the direct proof of theorem 2.3.In the third section, we get the extremal graph of weighted graph whose connectivity is 2.Theorem 3.3 Suppose G be weighted graph whose connectivity is 2, m, either G is Hamiltonian or through each vertex y of G there passes a cycle of weighted m. Furthermore, suppose and some vertex is contained in no cycle of length m .Then(1) there exists two vertecies u, v, the connected components C1 and C2 of G-u-v. are complete graphs;In the fourth section , we study G which satisfies the following two conditions In every triangle of G, either the three edges have different weights or the three edges have the same weight. Then G contains either a cycle of weighted at least m or a Hamiltonian cycle.In [5], Theorem 4.1 about a 173 type condition is given.Theorem 4.1 Suppose G be 2 connected weighted graph which satisfies and m. Then either G contains Hamilto...