There exists a 5-regular graph of size 50 such that the wheel graph W4 is an induced subgraph. T/f?
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There exists a 5-regular graph of size 50 such that the wheel graph W4 is an induced subgraph.
T/f?
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- Prove that in any simple graph G (i.e. no multiple edges, no loops) there exist two distinct vertices with the same degrees.Is there a 3-regular graph with order 5? Is there a 4-regular graph with order 5? Ifyes, draw such a graph, if no state why.Prove that in each component of the context of any graph, the number of vertices of an odd degree is always even.
- if graph G contains a cycle C, then every MST for G contains some edge of C. How do you prove this?(a) Draw a connected 3-regular graph with 6 vertices and no loops (b) Show that every 3-regular graph must have an even number of vertices. (c) Find two non-isomorphic connected 3-regular graphs with 6 vertices.Given an undirected graph G = (V.E), determine whether G contains a cycle of oddlength as subgraph.
- Say that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges (u, v), (v, w), (w, t). Show that a graph G with 99 vertices and no path of length three has at most 99 edges.If a complete bipartite graph Kr,s is regular what can you say about r and s?Prove or disprove that a graph is bipartite if and only if every induced cycle has even length.
- It is stated that K6 is not a planar graph (because K5 is one of the subgraphs of K6), but there is a subgraph of K6 which is a planar graph. If G' is a subgraph of K6, then find G' which has the maximum number of vertices and edges. Hint: Answers are expressed in sets and pictures.Prove that when doing depth-first search on undirected graphs thereare never any forward or cross edges.Consider a connected, undirected graph G with n vertices and m edges. The graph G has a unique cycle of length k (3 <= k <= n). Prove that the graph G must contain at least k vertices of degree 2.