- #1
Dragonfall
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Homework Statement
Show using a probabilistic method that a graph with average degree d has a stable set of cardinality at least n/(2d).
I can't think of a probabilistic method that will do this.
The stable set size in a graph refers to the maximum number of vertices in the graph that do not share any edges with each other. In other words, it is the size of the largest subset of vertices that do not form a complete subgraph.
The average degree of a graph, denoted by d, is the average number of edges incident to each vertex in the graph. It has been proven that the stable set size in a graph is at most floor(n/(d+1)), where n is the total number of vertices in the graph.
Understanding the stable set size in a graph can provide valuable insights into the structure and properties of the graph. It can also help in solving various optimization problems related to graph theory.
There are various techniques used to prove stable set size in graphs, including linear programming, semidefinite programming, and combinatorial optimization. These techniques involve formulating the problem as a mathematical optimization problem and finding the optimal solution.
Yes, the concept of stable set size in graphs has various real-world applications, such as in computer science, social networks, and logistics. For example, in computer science, it can be used to optimize data storage by finding the maximum number of non-overlapping tasks that can be executed simultaneously. In social networks, it can help in identifying groups of people with no connections to each other. In logistics, it can assist in optimizing transportation routes by finding the maximum number of locations that can be visited without any overlap.