Exemple de solution

G=hV, Eik≥1n=|V|
m=|E|
PV
k={pv
i|v∈V1≤i≤k}
pv
iv i
G k
ϕ
PV
kG=hV, Eik ϕ(G, k) = ϕ1∧ϕ2
ϕ1=V{u,v}∈EV1≤i≤k¬pu
i∨ ¬pv i;
ϕ2=Vv∈VW1≤i≤kpv
i
=Vv∈Vpv
1∨pv
2∨ · · · ∨ pv
k.
ϕ1ϕ2ϕ1m·k
ϕ2n k ϕ3
G=hV, Eik≥1ϕ(G, k)
G k
G=hV, Eik≥1
ϕ(G, k)G k
µ:PV
k→ {vrai,faux}µ|=ϕ(G, k)
µ|=ϕ2v∈V
1≤i≤k pv
iµ(pv i) = vrai
f:V→ {1,...,k}v∈V f(v)i µ(pv
i) = vrai
µ|=ϕ1{u, v} ∈ E1≤i≤k
µ(pu
i) = vrai µ(pv i) = faux f(u) = i f (v)6=i f
G f k G
k

G k ϕ(G, k)
f:V→ {1,...,k}G f
µ:PV
k→ {vrai,faux}v∈V1≤i≤k
µ(pv
i) = vrai f(v) = i
faux
µ|=ϕ(G, k)µ|=ϕ1µ|=ϕ2f
f(u)6=f(v){u, v} ∈ E i
f(u) = i f(v) = i µ(pu
i) = faux µ(pv i) = faux
µ|=ϕ1µ(pv
f(v)) = vrai v∈V µ |=ϕ2
χGG
1≤χG≤n
k k′k′≥k
χG∈[a, b]G c
c∈[a, b]χG∈[a, c]G c
χG∈[c, b]G c c =a+b
2