Heavy cycles in weighted graphs and heterochromatic cycles in colored graphs
Shenggui Zhang (Northwestern Polytechnical University, China)
Wednesday 10th July, 2013 16:00-17:00 Maths 516
In this talk, after an introduction of some terminology and notation of graph theory, we will present our new results on the existence of heavy cycles in weighted graphs and heterochromatic cycles in colored graphs. A weighted graph is one in which each edge is assigned a nonnegative real number, called the weight of the edge. The weight of a subgraph is defined as the sum of the weights its edges. An unweighted graph can be though as a weighted graph with weight one for each edge. The weighted degree of a vertex is the sum of the weights of the edges incident to it. Here we give several weighted degree conditions for the existence of heavy cycles in weighted graphs, which generalize previous results on the existence of long cycles in unweighted graphs. Colored graphs are defined similarly to weighted graphs, with colors associated with edges instead of weights. The color degree of a vertex in a colored graph is the number of colors assigned on the edges incident with it. We prove a color degree condition for the existence of heterochromatic triangles. The result conforms a conjecture of Li et al. Related problems are also posed.