(a) Let L be a minimum edge-cut in a connected graph G with at least two vertices. Prove that G − L has exactly two components.
(b) Let G an eulerian graph. Prove that λ(G) is even.
(a) Let L be a minimum edge-cut in a connected graph G with at least two vertices. Prove that G −...
49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is connected. b. Show that the result in (a) is best possible; that is, for each n 2 2, prove there is a graph with ("2) edges that is not connected. 49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is...
A connected simple graph G has 16 vertices and 117 edges. Prove G is Hamiltonian and prove G is not Eulerian
6. Prove that the following graphs are connected: (a) The 3 vertex cycle: (b) The following 4 vertex graph: (c) K 7. An edge e of a connected graph G is called a cut edge if the graph G obtained by deleting that edge (V(G) V(G) and E(G) E(G) \<ej) is not connected. Prove that if G1 and G2 are connected simple graphs which are isomorphic and if G1 has a cut edge, then G2 also has a cut edge....
Let G be a connected graph which is regular of degreer. Prove that the line graph of G, L(G), is Eulerian.
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.
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...
A.) Prove that if some graph G is an Eulerian graph, the L(G) {the line graph of G} is also Eulerian. B.) Find a connected non-Eulerian graph for which the line graph is Eulerian.
Let G be a simple graph with at least four vertices. a) Give an example to show that G can contain a closed Eulerian trail, but not a Hamiltonian cycle. b) Give an example to show that G can contain a closed Hamiltonian cycle, but not a Eulerian trail.
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)...
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