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.

Hamiltonian clets / bath in Bipartitegraphe Show that the following graph G is bipartite by partitioning the vertices into red (R)and blue (B)sets. No need to re draw e 9 Red set: Blue set: i) Does G have a Hamiltonian circuit? ii) Does G have a Hamiltonian path? ii) For parts (i) and (ii) show a circuit or a path if it exists. If there is none, justify why not.

Let G be the graph on vertices V1, V2, ..., V8 with the following adjacency matrix: /01000001 1 0 0 10000 100 100 1 000001 1 0 00 100 000 0001000 0001 0000 1 1 00000 i. Use the connectedness algorithm to find the connected component of v1. ii. Sketch all the graphs (up to isomorphism) on four vertices that are isomorphic to an induced subgraph of G (identifying a suitable induced subgraph in each case). iii. Sketch all the graphs (up to isomorphism) on four vertices that are not iso- morphic to an induced subgraph of G. iv. Find a graph on four vertices that is isomorphic to a subgraph, but not to an induced subgraph, of G. Explain your answer.

5. Let G = (V, E) be a graph with vertex-set V = {1,2,3,4} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2,4)}. (a) Draw the graph. Find (b) maximal degree, i.e. A(G), (c) minimal degree, i.e. 8(G), (d) the size of biggest clique, i.e. w(G), (e) the size of biggest independent set, i.e. a(G), ter (f) the minimal number of colours needed to color the graph, i.e. x(G).