The three questions are stated in an attached .doc file (1.doc).

In part 1, a diagram is given, and whether the object depicted in the diagram is a graph is to be determined.

In part 2, a graph is given and its chromatic number is to be determined.

In part 3, a graph is given and the number of colors needed to color the regions of the graph (to ensure that adjacent regions have different colors) is to be determined.

Complete solutions are given in an attached .doc file (1-Solution.doc).

In part 1, it is determined that the objected depicted in the given diagram is a graph; its vertex set and edge set are presented.

In part 2, the definition of chromatic ...

Solution Summary

The question of whether the object depicted in the given diagram in part 1 is a graph is answered. A detailed determination of the chromatic number of the graph in part 2 is presented, as is a detailed determination of the number of colors needed in part 3 (to ensure that adjacent regions have different colors). The definition of chromatic number of a graph is reviewed.

... The chromatic number of a graph is the least number of colors required to do coloring of that graph or more clearly we can say that the chromatic number of a ...

... easy to see that, for every n, the chromatic number of Gn ... it is impossible to 3-color such a graph (but it ... by way of contradiction, that there is a 3-coloring. ...

... to the last vertex) that contains every vertex in the graph. ... w in B . (e) λ(G) denotes the chromatic number of G ... edge, so we could use the same color for all ...

... The chromatic number of a graph is the least number of colours it takes to color its vertices so that adjacent vertices have different colours. ...

Chromatic Numbers and Graph Coloring. ... vertex chromatic number, or simply the chromatic number of , and is denote by , this problem is about graph coloring). ...

... collection of its faces, which are to be colored so that no adjacent faces have the same color.]. ... Then dual graph of G., ie G* having chromatic number (G)= 2 ...

... the chromatic number of G_1 + G_2 is X(G_1) + X(G_2) for any two graphs G_1 and G_2. Where X(G)is the Chromatic Number. defn: X(G) a proper colouring or simply ...

List-Chromatic Numbers. ... Please can you explain what does list-chromatic number means and don ... In a proper graph coloring, all vertices are colored such that any ...

... We know that the vertices 1,3,6,9,12 form a clique with size 5. So, we know that the chromatic number is at least 5. We can color the graph G in 5 ...