Graph Theory: Lecture No. 18
Kuratowsky’s Theorem: The following
assertions are equivalent for graphs G:
(1) Gis planar
(2) Gcontains neither K5nor K3,3as a
topological minor.
(3) Gcontains neither K3,3or K5as a minor
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Graph Theory: Lecture No. 18 - Kuratowski's Theorem and Graph Coloring, Slides of Design and Analysis of Algorithms
Lecture no. 18 of graph theory, discussing kuratowski's theorem, graph coloring, and the relationship between the choice number and chromatic number of a graph. The lecture explains that a graph is planar if and only if it does not contain k3,3 or k5 as a minor, and introduces the concept of k-list colorable or k-choosable graphs.
Typology: Slides
2012/2013
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Kuratowsky’s Theorem: The following assertions are equivalent for graphs G : (1) G is planar (2) G contains neither K 5 nor K 3 , 3 as a topological minor. (3) G contains neither K 3 , 3 or K 5 as a minor
Let G = (V , E ) and Sv , v ∈ V a family of sets. We call a vertex coloring c(v ) ∈ Sv for all v ∈ V a colouring from the lists Sv. The graph G is called k-list colourable or k-choosable if for every family (Sv ), v ∈ V with |Sv | = k for all v ∈ V , there is a (proper) vertex coloring of G. The least integer k for which G is k-choosable is called the choice number of G (or the list chromatic number) of G.
Every planar graph is 5 -choosable.
Suppose that every inner face of G is bounded by a triangle and its outer face by a cycle C = {v 1 , v 2 ,... , vk , v 1 }. Suppose further that v 1 has already been coloured with the colour 1 , and v 2 has been coloured with 2. Suppose finally that with every other vertex of C a list of at least 3 colours is associated and with every vertex of G − C a list of at least 5 colours. Then the colouring of v 1 and v 2 can be extended to a colouring of G from the given lists.