Discrete Mathematics [Lecture notes] by Gasper Fijavz

By Gasper Fijavz

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10 the expression 2|E(G)| ≤ 6|V (G)| − 12 = 6n5 + 6n6+ − 12. This implies that n5 ≥ 12. A graph G is called k-degenerate if we can obtain an empty graph by repetitively deleting vertices of degree ≤ k starting from G. Equivalently stated, every subgraph of G has a vertex of degree ≤ k. 1 is that planar graphs are 5-degenerate. Starting with a planar graph, deleting vertices of degree ≤ 5 will ultimately lead to the empty graph. 2 Let G be a planar graph with n vertices. Then min{deg(u), deg(v)} ≤ 10|E(G)| = O(n) e=uv∈E(G) Proof.

Observe that every edge appears exactly twice in the facial walks, once in each direction. 32 DM notes, c Gaˇsper Fijavˇz Similarly, given a collection of facial walks (a collection of walks in which every edge exactly twice, once in each of the directions) determines the rotation system: on one hand we can determine the lists of neighbors of each vertex v, on the other hand we can for each neighbor u of v determine its successor in the rotation system — we only need to find the vertex u which immediately precedes v, u in the collection of facial walks.

9 (Euler formula) Let G be a connected plane graph. Then |V (G)| − |E(G)| + |F (G)| = 2 35 DM notes, c Gaˇsper Fijavˇz Proof. Choose a spanning tree T of G. 2) in other words, Euler formula holds for the spanning tree T . Now adding an edge of E(G) \ E(T ) has the following effect: number of vertices does obviously not change, and the number of edges increases by one. An addition of a new edge (it is not a cutedge as together with T it forms a cycle) divides an existing face into a pair of new faces, and thus increases the number of faces by one as well.

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