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**Example text**

Therefore, the first digits after the decimal point √ in the decimal representation of b 6 must be 9. Since a990 ∈ N, the same is true for 990 √ √ √ ( 2 + 3)1980 = a990 + b990 6, so the answer is “9”. 2 Exponential Generating Functions ˆ Definition 10 For a sequence (an )n∈N we call A(z) = generating function of (an )n∈N . an n n≥0 n! n )n∈N , so we can make use of all results for the generating functions. n z n of the sequence (cn )n∈N with cn = n! n ak bn−k k! (n − k)! n )n∈N . ˆ ˆ B(z) ˆ Therefore, C(z) = A(z) holds if and only if for all n ∈ N: n n ak bn−k .

The path P cannot end in w enden, because in that case its last edge was an n(v)-edge and P together with the edge {v, w} would be a cycle of odd length (in contradiction to the assumption that G is bipartite). Hence, we can swap the colours n(v) and n(w) on P and colour the edge {v, w} with the colour n(w). ✷ Notation: Let G be a graph. For k ∈ N, we call G k-regular if all vertices in g are of degree k. We call an edge e ∈ E(G) a bridge if G − e contains more connected components than G. Theorem 46 Let G be a 3-regular planar graph without bridges.

Proposition 32 Let G be a graph with m edges. Then χ(G) ≤ 1 + 2 1 2m + . 4 Proof: In a colouring with χ(G) colours there must be an edge between each pair of colour classes (otherwise we could use the same colour for both classes). Thus m ≥ χ(G) = 2 1 χ(G)(χ(G) − 1), which is equivalent to the inequality of the proposition. ✷ 2 33 ¯ of a graph G is the graph that is defined by the Definition 12 The complement G ¯ ¯ ¯ := V (G) vertex set V (G) := V (G) and the edge set E(G) \ E(G). 2 Proposition 33 (Nordhaus and Gaddum [1956]) For every graph G with |V (G)| = n we have: √ ¯ ≤ n + 1, (a) 2 n ≤ χ(G) + χ(G) (b) n ¯ ≤ χ(G)χ(G) ≤ n+1 2 2 .