图论模型选讲

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1、K¨onigsbergNPC§1.1.1G V(G)E(G)V(G)V(G)

2、1G=V(G)={1,2,3,4,5}E(G)={(1,2),(2,3),(3,4),(1,3),(3,5),(3,5),(2,2),(1,5)}1234151-1(1)!"#$"&"(")2-#(2)ek=(vi,vj)vivj$!$"
e

3、k1(2,2)%
!$&"
(")"(1(2,2)(3,5)(1(3)!vivj$
"!eiej!"$
"ekvi#viek$(4)$()!%ek=vivj*&'&'(')G1=,G2=

4、()!,()'f:V1→V2,*(-vi,vj∈V1,(vi,vj)∈E1(∈E1)((f(vi),f(vj))∈E2(∈E2)/(vi,vj)()(f(vi),f(vj))(∼=G)"
G1G2)*G122+",+*-+,!Peterson1A6FGB52J107HI98EC43D1-21.20v∈V(G),v0d(v)=dG(v)

5、Gv1,23!d(v)=0v-d(v)=1v-.!/-.!d(v)"
v"!d(v)#
v#$()-00.&2&41G=
d(v)=2

6、E

7、0+25v∈V1#6"1.3HG)H⊆G!V(H)⊆V(G)E(H)⊆E(G)HG)H⊂G!H⊆GH=GHG7!H⊆GV(H)=V(G)⊆V

8、G=VE⊆EVG)G[V]VG$&V##E)G[E]EGE##21.4/2GG49%P=v0e1v1e2···elvl:v0)vl/2vr−1vrer$r=0,1,···,lv0,vl*P0/PlP;0!v0=vl
P

9、=vl
P<!P&50
P!6v0=vl
P1&17'=>? 12;1(#;22:u)w/2((u,w)-/2)6:u)w(u,w)-!+*&,
/x,y()d(x,y)xy+*;0/1.5##34(1)5):6$-"+*& 5)$?():6(2)

10、G=!V(G);#XY*(E(G)$&XY2"!(;XY)x∈Xy∈Y&(x,y)
5)2G⇐⇒G"#