Is this statement about the chromatic number true? Please give some prove and brief explaination: If A and B are graphs on the same vertex-set, then χ(A ∪ B) ⩽ χ(A) + χ(B).

Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.

Please solve with the computer Question 2:  Draw a simple undirected graph G that has 11 vertices, 7 edges.

The vertex set of graph G is {a, b}. There is one undirected edge {a, b} in G. (a) Does G have a Hamiltonian cycle? Justify your answer.

Consider an undirected graph G with 100 nodes. The maximum number of edges to be included in G so that the graph is not connected is

Prove that the two graphs below are isomorphic.

Is there a connected planar graph with an odd number of faces where every vertex has degree 6? Prove your answer.

Prove that in any simple graph G (i.e. no multiple edges, no loops) there exist two distinct vertices with the same degrees.

• If a planar graph has 11 vertices and four times as many edges as faces, how many faces does it have? Is there a planar graph with 13 vertices that has three times more edges than faces?

A graph is biconnected if every pair of vertices is connectedby two disjoint paths. An articulation point in a connected graph is a vertex that woulddisconnect the graph if it (and its adjacent edges) were removed. Prove that any graphwith no articulation points is biconnected. Hint : Given a pair of vertices s and t and apath connecting them, use the fact that none of the vertices on the path are articulationpoints to construct two disjoint paths connecting s and t.