A tournament is a simple directed graph such that if u and v are distinct vertices in the graph, exactly one of (u, v) and (v, u) is an edge in the graph. Prove that every tournament has a Hamilton path.
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A tournament is a special kind of directed graph where for every pair of vertices \(u\) and \(v\), there is exactly one directed edge between them, either from \(u\) to \(v\) or from \(v\) to \(u\). A Hamilton path in a graph is a path that visits each vertex Show more…
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A tournament is a complete oriented graph, that is, a directed graph in which for any two distinct vertices x and y either there is an edge from x to y or there is an edge from y to x, but not both. Prove that every tournament has a directed Hamilton path.
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A simple graph is a finite set V of vertices together with a set E of edges. An edge is a two-vertex set {u,v} that can be represented as a line from u to v. The degree of a vertex v is the number of edges containing v. For example, the diagram shows a simple graph with eight edges and six vertices with degrees ranging from 0 to 4. Prove that any simple graph with n ≥ 2 vertices has at least two vertices of equal degree.
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