or Xglqt G qcq ipcq.es qq.TO ve E 2 Vz Case I X Cuz XCuz 0 g - - PDF document

SLIDE 1

Colorings

Chromatic

Polynomial

DeletionContraction Formula Stanley SymmetricChromaticFunction

specialization Symmetry

Deletion Contraction

Coloringof a graph G

X VG

1,2

9222

17

Ppezr coloring HedgesLynde

EGS

01

022

XLV tXCqj

22 1

Howto count propercolorings

ChromaticPolynomiat

g I

g I

G

• r Xglqt

qcq

ipcq.es

qq.TO

Vz

ve E2

CaseI

X

Cuz XCuz

g10

gI

qlqDLq.cl

H

XH g

case

xcua xcv

v0

0 1 1

9cg174

234

2

q

3

qCq1 Cq1

qCq1

g2 Cq2

Special cases

Ed

RecursiveMethod Deletion Contraction

000

02

qtr

in

Anyproper coloringof G

isalso

G

a proper coloring

f e

i

true

However proper colorings of G e

include cases wherexcv xw

SLIDE 2

include cases hereXcv

Xwa These are colorings when y Vz

Cecontracted

Xdg XGe.GL Xue q

11

66

ft

du du

fo

9

• O

O

k b

dy

ga

• r

O

O

G O

O

O O

• f

q2

q2 9

If

g

q3q2 q2q q2q

q3 3q2t2q

q q 1Cq2

Stanley SymmetricChromatic Function

Infinite colors

eachcolor

gets 1 variable

Xl Xz etc

Xdr xz

I

xfzpert.IE

Xo

x xz D

xtxxj x.xxyix.MXI

A

t XXzXzt XzXyt

XLI

A

XZX

Xg

1 2 3

X tXzXzXyt

2 XixXk

i j k

i k

Symmetric

canswitcharoundvariablesandnothing

changes

a sX

SLIDE 3

Specialization for some n let Xu

f EET

n

f

I

Xun

1

I l l

I 0,0

n

Symmetricbasis Powerfunction

basisPk

Pk

X

• k

P _41 21

Ps Xp

124

X

X Xc

I

xixjxki

j

tk.it

k

P

3p

Pz 2ps

n3 3h42m

XiXjXk

3XiXjXj

3XiXjXj

1 Xi

Xi Xi 13Xi Xi Xi

Nole

specialization

E 1 for Ken is

preciselypreen

Pk xYtxEtxYt

147kt1kt

140kt

i

n n