Let k be an integer satisfying k ≥ 2. Let G be a connected graph with...

Let k be an integer satisfying k ≥ 2. Let G be a connected graph with no cycles and k vertices. Prove that G has at least 2 vertices of degree equal to 1.

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Colby Messinger answered 1 year ago

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G...

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G has no cut-edge. (Hint: Use the bipartite version of handshaking.) b) Construct a simple, connected, nonbipartite 3-regular graph with a cut-edge. (This shows that the condition “bipartite” really is necessary in (a).) 2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is the number of spanning trees in F_n. Use deletion/contraction to prove that a_n = 3a_n-1 - a_n-2...

Let G be a connected graph and let e be a cut edge in G. Let...

Let G be a connected graph and let e be a cut edge in G. Let K be the subgraph of G defined by: V(K) = V(G) and E(K) = E(G) - {e} Prove that K has exactly two connected components. First prove that e cannot be a loop. Thus the endpoint set of e is of the form {v,w}, where v ≠ w. If ṽ∈V(K), prove that either there is a path in K from v to ṽ, or...

13.6 Let G be a simple connected cubic plane graph, and let pk be the number...

13.6 Let G be a simple connected cubic plane graph, and let pk be the number of k-sided faces. By counting the number of vertices and edges of G, prove that 3p3 + 2p4 + p5 - c7 - 2p8 - • • • = 12. Deduce that G has at least one face bounded by at most five edges.

Let x be a fixed positive integer. Is it possible to have a graph G with...

Let x be a fixed positive integer. Is it possible to have a graph G with 4x + 1 vertices such that G has a vertex of degree d for all d = 1, 2, ..., 4x + 1? Justify your answer. (Note: The graph G does not need to be simple.)

Graph Theory: Let S be a set of three pairwise-nonadjacent edges in a 3-connected graph G....

Graph Theory: Let S be a set of three pairwise-nonadjacent edges in a 3-connected graph G. Show that there is a cycle in G containing all three edges of S unless S is an edge-cut of G

4. Let n ≥ 8 be an even integer and let k be an integer with...

4. Let n ≥ 8 be an even integer and let k be an integer with 2 ≤ k ≤ n/2. Consider k-element subsets of the set S = {1, 2, . . . , n}. How many such subsets contain at least two even numbers?

Let G be a connected graph which contains two spanning trees T1 and T2 that do...

Let G be a connected graph which contains two spanning trees T1 and T2 that do not share any edges (note that G may contain edges that are in neither trees). Prove that G does not have a bridge. I have a hint for the answer: Show that each edge is in a cycle (2 cases). But I can't figure out the 2 cases. Some help please!

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Let k be an integer satisfying k ≥ 2. Let G be a connected graph with...

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