SLIDE 1
Colouring
Colouring For a graph G = (V, E), a colouring is a function c: V → N such that uv ∈ E implies c(u) = c(v). A graph is k-colourable if there is a colouring c which uses at most k colours, i. e., c: V → {1, . . . , k}.
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Colouring Colouring Colouring For a graph G = ( V , E ) , a - - PowerPoint PPT Presentation
Colouring Colouring Colouring For a graph G = ( V , E ) , a - - PowerPoint PPT Presentation
Colouring Colouring Colouring For a graph G = ( V , E ) , a colouring is a function c : V N such that uv E implies c ( u ) = c ( v ) . A graph is k -colourable if there is a colouring c which uses at most k colours, i. e., c : V {
SLIDE 2
SLIDE 3
Colouring
The k-Colouring problem asks, for a given graph, it is k-colourable. Theorem There is (probably) no polynomial time algorithm to determine if a given graph is k-colourable for all k ≥ 3. Some observations
◮ G is k-colourable if and only if each block of G is k-colourable. ◮ If G has a clique of size k, a colouring for G requires at least k colours.
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SLIDE 4
Colouring
Theorem The 2-colouring problem can be solved in linear time. 2-Colouring Algorithm
◮ Pick an arbitrary vertex v. ◮ Run a BFS starting in v. ◮ For all vertices u in even distance to v, set c(u) := 1. For all
vertices u in odd distance to v, set c(u) := 2.
◮ If there is an edge uw with c(u) = c(w), then G is not 2-colourable.
Graphs which are 2-colourable, are also called bipartite.
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SLIDE 5
Colouring Planar Graphs
SLIDE 6
Colouring Planar Graphs
Recall: Theorem For a connected planar graph G = (V, E) with at least three vertices, |E| ≤ 3|V| − 6. Therefore: Theorem Each planar graph contains a vertex with degree at most 5.
- Proof. By pigeon-hole principle.
- Thus, there is a vertex order v1, v2, . . . , vn such that vi has degree at
most 5 in Gi = G[{vi, vi+1, . . . , vn}].
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SLIDE 7
Colouring Planar Graphs
Input: A planar graph G. Output: A colouring c: V → N+.
1 Find a vertex order v1, v2, . . . , vn such that vi has degree at most 5
in Gi.
2 For i := n downto 1 3
Assign c(vi) the lowest available colour greater or equal to 1. Theorem There is a linear time algorithm to find a 5-colouring and an O(n2) time algorithm to find a 4-colouring for planar graphs. There is (probably) no polynomial time algorithm to find a 3-colouring for a planar graph, even it is known that such a colouring exist.
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