CSC 1300 – Discrete Structures 17:

Major Themes • coloring • Chromac number χ(G) Graph Coloring • Map coloring • Greedy coloring • CSC 1300 – Discrete Structures Applicaons Villanova University

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What is the least number of colors needed for the verces of Vertex Colorings this graph so that no two adjacent verces have the same color? Adjacent verces cannot have the same color χ(G) =

Chromac number χ(G) = least number of colors needed to color the verces of a graph so that no two adjacent verces are assigned the same color?

Source: “Discrete Mathemacs” by Chartrand & Zhang, 2011, Waveland Press. Source: “Discrete Mathemacs with Ducks” by Sara-Marie Belcastro, 2012, CRC Press, Fig 13.1. 4 5

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Map Coloring Map Coloring What is the least number of colors needed to color a map? Region è vertex Common border è edge

G A D C E B F G H I

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Coloring the USA Four color theorem Every planar graph is 4-colorable

The proof of this theorem is one of the most famous and controversial proofs in mathemacs, because it relies on a computer program. It was first presented in 1976. A more recent reformulaon can be found in this arcle:

Formal Proof – The Four Color Theorem, Georges Gonthier, Noces of the American Mathemacal Society, December 2008.

hp://www.printco.com/pages/State%20Map%20Requirements/USA-colored-12-x-8.gif hp://www.ams.org/noces/200811/tx081101382p.pdf

hp://people.math.gatech.edu/~thomas/FC/usa.gif

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Four color theorem Four color theorem

Every planar graph is 4-colorable Every planar graph is 4-colorable Do you always need four colors? What about non-planar graphs?

K3,3 K5

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Example Example

Source: “Discrete Mathemacs with Ducks” by Sara-Marie Belcastro, 2012, CRC Press, Fig 13.1. Villanova CSC 1300 - Dr Papalaskari 15 Villanova CSC 1300 - Dr Papalaskari 16

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Chromac Numbers of Some Graphs Example • χ(G) = 1 iff ...

• For Kn, the with n verces, χ(Kn) =

Corollary: If a graph has Kn as its subgraph, then χ(Kn)=

• For Cn, the cycle with n verces, χ(Cn) =

• For any biparte graph G, χ(G) =

• For any planar graph G, χ(G) ≤ 4 (Four Color Theorem)

Source: “Discrete Mathemacs with Ducks” by Sara-Marie Belcastro, 2012, CRC Press, Fig 13.1. Villanova CSC 1300 - Dr Papalaskari 17 Villanova CSC 1300 - Dr Papalaskari 18

Applicaons of Graph Coloring Example: Schedule these exams, avoiding conflicts • map coloring • scheduling Monday Tuesday Wednesday – eg: Final exam scheduling

• Frequency assignments for radio staons CSC 2053 CSC 1052 CSC 1300 CSC 2400 CSC 4480 • Index register assignments in compiler CSC 1700 opmizaon CSC 2014 • Phases for traffic lights

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Earlier example – seen as scheduling constraints Revised Exam Schedule:

CSC 2053 CSC 1700 Monday Tuesday Wednesday CSC 1700 ?? CSC 1052

CSC 2014 CSC 2053 CSC 2400 CSC 4480 CSC 1300 CSC 4480

CSC 2014

CSC 2400 CSC 1300 CSC 1052

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Compung the Chromac Number Graph coloring algorithm?

There is no efficient algorithm for finding χ(G) for arbitrary graphs. Most computer sciensts believe that no such algorithm exists.

Greedy algorithm: sequenal coloring: 1. Order the verces in nonincreasing order of their degrees. 2. Scan the list to color each vertex in the first available color, i.e., the first color not used for coloring any vertex adjacent to it.

Not always opmal! (order maers)

Source: “Discrete Mathemacs with Ducks” by Sara-Marie Belcastro, 2012, CRC Press, p374.

Villanova CSC 1300 - Dr Papalaskari 24 hp://upload.wikimedia.org/wikipedia/commons/0/00/Greedy_colourings.svgVillanova CSC 1300 - Dr Papalaskari 25

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Another Applicaon of vertex coloring: Example: Index Registers Traffic lights

• see also example 13.3.9 & Figure 13.12

source: hp://www.lighterra.com/papers/graphcoloring/

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Another Applicaon of vertex coloring: Traffic lights

• see also example 13.3.9 & Figure 13.12

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