Prove that in every simple graph there is a path from every vertex of odd degree to a vertex of odd degree.
Prove that in every simple graph there is a path from every vertex of odd degree...
Exercise 5. Let G be a graph in which every vertex has degree at least m. Prove that there is a simple path (i.e. no repeated vertices) in G of length m.
Prove that, if G is a nontrivial connected graph in which every vertex has even degree, then the edge connectivity of G is no less than 2. Prove that, if G is a nontrivial connected graph in which every vertex has even degree, then the edge connectivity of G is no less than 2.
An odd graph is one where each vertex is of odd degree. Show that a graph is odd if and only if a(X) = |x|(mod2) for each subset X of V.
Prove that, if even degree, then the edge conn G is a nontrivial connected graph in which every vertex has Prove that, if even degree, then the edge conn G is a nontrivial connected graph in which every vertex has
Prove that a tree with at least two vertices must have at least one vertex of odd degree.
Say that we have an undirected graph G(V, E) and a pair of vertices s, t and a vertex v that we call a a desired middle vertex . We wish to find out if there exists a simple path (every vertex appears at most once) from s to t that goes via v. Create a flow network by making v a source. Add a new vertex Z as a sink. Join s, t with two directed edges of capacity...
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2. Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
7. Prove that for every tournament T, there exists some vertex s for which there is a directed path from s to r of length at most 2 for every vertex r EV(T)
8. This question has two parts. (i) Let G be a graph with minimum vertex degree 8(G) > k. Then prove that G has a path of length at least k. (ii) Let G be a graph of order n. If S(G) > nz?, then prove that G is connected.
3. Prove that every connected graph has a vertex whose removal (including all adjacent edges will not disconnect the graph by describing a DFS method that finds such a vertex.