1: Prove the directed version of Veblen's Theorem : A directed graph admits a decomposition into directed cycles if and only if its even.  2: Show that a graph G is connected if and only if there is an (X,Y) - path in G for any two nonempty subsets X and Y of V.

Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only if u and v are in the same connected component of G.  

Let G = (V, E) be a graph with vertex-set V = {1, 2, 3, 4, 5} and edge-setE = {(1, 2),(3, 2),(4, 3),(1, 4),(2, 4),(1, 3)}.(a) Draw the graph.Find  (b) maximal degree, i.e. ∆(G),(c) minimal degree, i.e. δ(G),(d) the size of biggest clique, i.e. ω(G),(e) the size of biggest independent set, i.e. α(G), ter(f) the minimal number of colours needed to color the graph, i.e. χ(G).

Let u and v be distinct vertices in a connected graph G. There may be several connected subgraphs of G containing u and v. What is the minimum size of a connected subgraph of G containing u and v? Explain your answer.

Prove that if a plane graph G has n vertices, e edges, f faces and k components,then n−e+ f = k +1.

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Let G be a simple connected graph.  (a) Suppose that G has t blocks. Prove that there is an ordering (B1,B2,...,Bt) of the blocks of G such that, for each i ∈ {2,3,...,t}, the graphs Bi and B1 ∪B2∪···∪Bi−1 have exactly one vertex in common. (b) Prove that the chromatic number of G is the maximum of the chromatic number of each block of G.

Consider the following directed graph. b Identify the correct statement about the antisymmetric property of the relation represented by the given directed graph. (You must provide an answer before moving to the next part.)

Let G be a connected graph that has exactly 4 vertices of odd degree: v1, V2, V3 and V4. Show that there are paths with no repeated edges from v1 to V2, and from v3 to v4, such that every edge in G is in exactly one of these paths.

Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane graph with n vertices and girth g.