By Bernhard Korte

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Extra info for Combinatorics, Graphs, Matroids [Lecture notes]

Sample text

N−k • For the derangement number Dn , we have already proved the formula n! = nk=0 nk Dk . )n∈N is the convolution of (Dn )n∈N and the sequence 1, 1, 1, . . 1 z n = ez . )n∈N , so n! ˆ D(z) · ez = z n = n! This implies zn = n≥0 e−z ˆ D(z) = . 1−z 32 1 . 1 (−1)n z n and 1−z = n≥0 z n as (standard) generating functions. By comparing coefficients in (5), this gives us once again the equation Dn = n! n k=0 1 (−1)k . k! II Graphs 5 Planar Graphs For the lectures about planarity of graphs we refer to the textbook by Korte and Vygen [2012].

2006]: The strong perfect graph theorem Annals of Mathematics, 164, 2006, 51–229. Diestel, R. [2005]: Graph Theory. Third edition, Springer, 2005. , Schrijver, A. [1984]: Polynomial algorithms for perfect graphs. Annals of Discrete Mathematics, 21, 1984, 325–356. , Jungnickel, D. [2004]: Einf¨ uhrung in die Kombinatorik. 2nd edition. De Gruyter, 2004. , Vygen, J. [2012]: Combinatorial Optimization. Theory and Algorithms. 54th edition. Springer, 2012. A. [2011]: Bijective Combinatorics. CRC Press, 2011.

Then choose a vertex v1 in G1 with colour c1 . There must be a vertex v2 in G2 with colour c2 = c1 because G2 cannot be coloured with just one colour. Since G3 is not 2-colourable there must be a vertex v3 ind G3 with colour c3 ∈ {c1 , c2 }. We can continue this and get for each i ∈ {1, . . , k − 1} a vertex vi in Gi whose colour ci is not contained in {c1 , . . , ci−1 }. But there is a vertex v ∈ Ak with τk (v) = (v1 , . . , vk−1 ). Thus, v is in Gk connected by an edge to all vertices in {v1 , .