Probability
# Graph Theory

Cosmic liaisons are established among the nine planets of the solar system. Rockets can only travel along the following routes:

Can a traveler get from Earth to Mars?

Five cities $P,Q,R,S,T$ are connected by different modes of transport as follows

- $P$ and $Q$ connected by boat as well as rail.
- $S$ and $R$ connected by bus and boat.
- $Q$ and $T$ connected by air only.
- $P$ and $R$ connected by boat only.
- $T$ and $R$ connected by rail and bus.

If a person visits each of the places starting from $P$ and gets back to $P,$ which of the following places must he visit **twice**?

What is the least number of colors needed to color each vertex of the graph below such that no two adjacent vertices have the same color?

Note: In graph theory, two vertices are adjacent if they are connected by an edge.

*also* to all friends of friends, then whose party will have the most invites?

**Notes and assumptions**

- Each dot represents a person.
- Each line represents mutual friendship between the people on either end .

**cannot** be traced without lifting up your pencil and without tracing over the same edge twice?