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s08.0

# s08.0 - Last time terminology reminder w Simple graph...

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Last time: terminology reminder w Simple graph Vertex = node Edge Degree Weight Neighbours Complete Dual Bipartite Planar Cycle Tree Path Circuit Components Spanning Tree

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vertex (node) edge Vertex degree is 3
Spanning Trees A spanning tree of a graph G is a tree that touches every node of G and uses only edges from G Every connected graph has a spanning tree

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Dual = put a node in every face, and an edge between every adjacent face Dual Graph
G ra ph C o lo uring

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Graph Colouring Graph Colouring Problem: G ive n a g ra ph, c o lo ur a ll the  ve rtic e s  s o  that two  a djac e nt ve rtic e s  g e t diffe re nt c o lo urs . Objective:  us e   m inim um  num b e r o f c o lo urs . 3-c o lo ura ble
Optimal Colouring What g raphs  have  c hro m atic  num b e r o ne ? whe n the re  a re  no  e dg e s … Wha t g ra phs  ha ve  c hro m a tic  num be r 2? A pa th?   A c yc le ?  A triang le ? What g raphs  have  c hro m a tic  num be r larg e r tha n 2? Definition. m in #c o lo rs  fo r G  is  chromatic number,  ( G )

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Simple Cycles e ve n (C 2 ) = χ od d 3 ) =
n n (K ) = χ Complete Graphs

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e ve n (W 3 ) = χ od d 4 ) = 5 W Wheels
Trees Pic k a ny ve rte x a s  “ro o t.” if (uniq ue ) pa th fro m  ro o t is o dd le ng th: ro o t C a n pro ve  m o re  fo rm a lly us ing  induc tio n (c las s wo rk).

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Chromatic Number Wha t g ra phs  are  3-c o lo urab le ? No  o ne  kno ws … Ho w do  we  e s tim a te  the  c hro m atic  num b e r o f a  g ra ph? If the re  is  a  c o m ple te  s ubg ra ph o f s ize  k,  the n we  ne e d a t le as t k c o lo urs ? YES Is  the  c o nve rs e  true ? If a  g ra ph has  no  c o m ple te  s ubg ra ph o f s ize  4,  the n we  c an c o lo ur it us ing  4 c o lo urs ?
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s08.0 - Last time terminology reminder w Simple graph...

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