There Are Additional Conditions Of Heavy Circle. Weighted Graph

This thesis deals with the topic of paths and cycles in weighted graphs. Here a weighted graph is a one in which each edge is assigned a non-negative number, called the weight of the edge. The weight of a path (cycle) is the sum of the weights of its edges. The weighted degree d^w(v) of a vertex v is the sum of the weights of the edges incident with the vertex.In the first chapter, after introducing some basic terminology and notations, we give a brief overview to the main results of the thesis.In Chapter 2, we get the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: (1) The weighted degree sum of any three pairwise nonadjacent vertices is at least m; (2) For each induced claw and each induced modified claw of G, all of its edges have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/3. This generalizes an early result of Zhang, Broersma and Li on the existence of heavy cycles in weighted graphs.In Chapter 3, we prove that: Suppose G is a k-connected weighted graph which satisfies the following conditions: (1) The weighted degree sum of any k + 1 pairwise nonadjacent vertices is at least m; (2) In each induced claw, each induced modified claw and each induced P₄ of G, all edges have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/(k + 1). This generalizes a result of Enomoto, Fujisawa and Ota on the existence of heavy cycles in k-connected weighted graphs.Enomoto, Fujisawa and Ota obtained some properties of the connected weighted graphs which satisfies the following conditions: (1) w(xz) = w(yz) for every vertex z N(x) N(y) with d(x,y) = 2; (2) In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. In Chapter 4, we obtain some properties of the connected weighted graphs which satisfies the condition (2) of the theorem we get in Chapter 3. With these properties, we give a simpler proof of a theorem of Zhang et al. and the theorem we get in Chapter 3.We conclude the thesis with Chapter 5 by proposing some problems for further research on the paths and cycles in weighted graphs.