Draw an (8, 15)-graph G with χ^, (G) = 5

Let G be a graph having no isolated vertex and no induced subgraph with exactly two edges. Show that G is a complete graph.

Check whether the sequence (4, 4, 4, 3, 2, 2, 1, 1, 1) is graphic or not. If yes, draw a graph realising this degree sequence.

Define a flow on the following network, having value at least 5.

Draw the dual of the following plane graph.

Does the dual have any cut-vertex? Justify.

If G is a k-connected graph having n vertices, what is the minimum size of G? Justify .

Every 3-colourable graph contains an odd cycle. True or false? Justify.

There exists a self-complementary graph on 2023 vertices. True of false? Justify your answer.

There exists a self-complementary graph on 2023 vertices. True of false? Justify your answer.

There exists a self-complementary graph on 2023 vertices. True of false? Justify your answer.

Draw a diagram, as nice as possible, of the line graph of the Petersen graph. Write the number of vertices, the number of edges, the minimum and maximum degrees of it.

Verify the K¨onig Egarv´ary Theorem for the following graph.

Draw the complement of the following graph.

Is the complement Hamiltonian? Justify your answer.

(a) An n-vertex forest with n/2 edges has exactly n/2 trees as its components. True or false? Justify.

Find a minimum-weigh spanning tree in the following graph

Does there exist a 3-edge-colourable graph on 10 vertices and having 20 edges? Justify.

There exists a complete binary tree on 15 vertices.

The line graph of the Petersen graph has 30 edges.

If κ(G) < κ0 (G), then δ(G) ≥ 4.

The complement of a disconnected graph is connected.