Question

1.  A planar graph is said quadrag enary if all the faces of the graph are
quadrilaterals (all faces are formed by 4 arcs). Let n be the number of
vertices of the graph.
 Find a formula that gives the number of arcs of a graph
quadrag enary according to n. Also find a formula that gives the number of
faces of a quadrag enary graph according to n.
You must clearly write your two formulas. You have to prove that your two
formulas are correct using Euler's formula.
2. 
(a) Let G = (V; E) be a non-oriented graph with V = fv1; v2; v3; v4; v5; v6; v7; v8; v9g and
deg (v1) = 0, deg (v2) = 1, deg (v3) = 1, deg (v4) = 1, deg (v5) = 2, deg (v6) = 2,
deg (v7) = 3, deg (v8) = 4 and deg (v9) = 6. Does such a graph exist? If yes,
draw it. If not, prove it.
(b) Let G = (V; E) be a non-oriented graph with V = fv1; v2; v3; v4; v5; v6; v7; v8; V9G,
where deg (v1) = 0, deg (v2) = 1, deg (v3) = 1, deg (v4) = 2, deg (v5) = 2, deg (v6) = 2,
deg (v7) = 3, deg (v8) = 4 and deg (v9) = 4. Does such a graph exist? If yes,
draw it. If not, prove it.

Answer 1

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