Suppose that an edge e ::= {v1, v2 } is on a cycle in a graph G where v1, v2 are vertices of G. If e is removed from G to create the graph G e, then there exists a path from v1 to v2 in G – e. If e is removed from G to create the graph G e, then there does not exist a path from v1 to v2 in G – e. O If e is removed from G to create the graph G - e, then it is not possible to determine if there exists a path from vị to v2 in G – e.

I have to prove the following Corollary: "Let u and v be vertices of a 2-connected graph G. Then there is a cycle of G that contains both u and v.”   My idea of the proof is the following: Given vertices u and v, I want to show that there is a cycle containing both. Then by contradiction, I could assume that there exists u and v sucht that there are no cycles C including u and v.  However, I'm not quite sure that this is the correct approach. I would appreciate some help to prove this corollary. Thank youuu

Let G be a graph and let e1 and e2 be edges.  Show that if deleting e1 and e2 disconnects vertices u and v, then any cycle containing both u and v must contain both e1 and e2.  One approach:  You could apply to the graphs G-e1 and G-e2 the fact that if e is a bridge whose removal disconnects u and v, then any path connecting u and v must contain e.

Let K, be a graph obtained from K, by deleting one edge. How many subgraphs of K, are isomorphic to K, ?

-nsider the following graph G(V, E) w • The set of vertices V: {2,3, 4, 6, 8, • The set of edges E: {(a, b) e Z² : « a) Draw the graph G. b) Draw the graph G3

8. If we remove an edge e from a graph G and G-e is still connected, then show that e lies along some cycle of G.

Let G be a connected graph of order n and size n. Prove that G contains a single cycle.

The symmetric difference graph of two graphs G. (V. E.) and G₂ (V₁ E₂) on the same vertex Set is defined as G₁ AG₂:= (V, E, DE₂). E₁ DE ₂ = (E₁ \ E₂) U (E₂\E.) : E₁ E₂ = €₁ 0 € ₂ If G₁ and 6₂ are euleran, show that every Vertex in G, D G₂ has even degree FACT: Each vertex of a evlenan graph has even degree. SO: Show G₁ D G₂ is eulerian.

a. Let V = {x, y, z, u, v}. How many distinct graphs are there on the vertex set V with exactly 6 edges? (Note that for graphs G₁ = (V, E₁) and G₂ = (V, E₂) to be distinct, they need only have E₁ # E2, but they may be isomorphic.) b. In the graph below, give an example of a trail that is not a path. I Y U Z