49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then...
Let G be a graph with n vertices and n edges. (a) Show that G has a cycle. (b) Use part (a) to prove that if G has n vertices, k components, and n − k + 1 edges, then G has a cycle.
PLEASE HELP Let G is a graph with 2n vertices and n^2 edges. An amicable pair of vertices is an unordered pair (u, v), such that dist(u, v) = 2. Prove that G has at least n(n − 1) amicable pairs of vertices.
Let G be a simple graph with 2n, n 2 vertices. Suppose there are at least n2 1 edges. Show that at least one triangle is formed. Hint: Check n 2 first and then use induction Let G be a simple graph with 2n, n 2 vertices. Suppose there are at least n2 1 edges. Show that at least one triangle is formed. Hint: Check n 2 first and then use induction
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G. (a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then...
Show that every connected graph with n vertices has at least n - 1 edges. (It can be done by induction, for example).
Most Edges. Prove that if a graph with n vertices has chromatic number n, then the graph has n(n-1) edges. Divide. Let V = {1, 2, ..., 10} and E = {(x, y) : x, y € V, x + y, , and a divides y}. Draw the directed graph with vertices V and directed edges E.
Prove that any graph with n vertices and at least n + k edges must have at least k + 1 cycles.
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is removed from C then the subgraph S C G that remains is still connected. "Directly' means using only the definitions of the concepts involved, in this case connected' and 'circuit'. Hint: If z and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y...
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge ={a, b} is removed from then the subgraph S CG that remains is still connected. Directly' means using only the definitions of the concepts involved, in this case 'connected' and 'circuit'. Hint: If r and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y that avoids e? (b)...
A connected simple graph G has 16 vertices and 117 edges. Prove G is Hamiltonian and prove G is not Eulerian