2014年11月21日，西北工业大学张胜贵教授为中心师生作题为“Heavy Cycles in Weighted Graphs and Heterochromatic Cycles in Colored Graphs”的学术报告。
A simple graph is called a weighted graph if each of its edges is associated with a nonnegative number, called the weight of the edge. The weight of a subgraph of a weighted graph is defined as the sum of the weights of its edges. The weighted degree of a vertex is defined as the sum of weights of the edges incident with it. In this talk, we will first give an overview of the results which give weighted degree conditions for the existence of heavy cycles (cycles with large weights) in weighted graphs. These results generalize or extend classical ones on the existences of long cycles in (unweighted) graphs. At the same time, we propose some related problems.
A colored graph is a simple graph in which every edge is associated with a color. The color number of a graph is the number of colors associated with the edges of the graph. The color degree of a vertex in a colored graph is defined as the number colors of the edges incident with it. A cycle is called rainbow if all of its edges have different colors. In this talk, we will introduce some new results on the existence of rainbow cycles of length 3 or 4 under the color number and color degree condition. Our results confirm a conjecture of Li et al. on the existence of rainbow triangles in colored graphs. We also propose some related problems.