Number of 4-contractible edges in 4 -connected graphs

Number of 4-contractible edges in 4 -connected graphs

Number of 4-contractible edges of 4-contractible edges 4-contr of 4-contractible edges actible edges in 4 -connected gr of 4-contractible edges aphs Yos himi Egawa (Tokyo Univer of 4-contractible edges s ity of 4-contractible edges Science) For of 4-contractible edges a gr of 4-contractible edges aph , is def 4-contractible edges 4-connected and is connected f 4-contractible edges or of 4-contractible edges ever of 4-contractible edges y with . def 4-contractible edges For of 4-contractible edges a 4-connected gr of 4-contractible edges aph and an edge , is 4-contr of 4-contractible edges actible

is 4-connected. ( ) { ( )| is 4 contr of 4-contractible edges actible } A char of 4-contractible edges acter of 4-contractible edges ization of 4-contractible edges 4-connected gr of 4-contractible edges aphs with was obtained by Fontet and by Mar of 4-contractible edges tinov. In par of 4-contractible edges ticular of 4-contractible edges , the f 4-contractible edges ollowing r of 4-contractible edges es ult is known. Fact A. If 4-contractible edges is a 4-connected gr of 4-contractible edges aph with , then G is 4-r of 4-contractible edges egular of 4-contractible edges . Bas ed on this f 4-contractible edges act, the f 4-contractible edges ollowing r of 4-contractible edges es ults wer of 4-contractible edges e obtained (her of 4-contractible edges e ). Fact A. If 4-contractible edges is a 4-connected gr of 4-contractible edges aph with , then is 4-r of 4-contractible edges egular of 4-contractible edges . Theor of 4-contractible edges em B (Ando & E.).

If 4-contractible edges is a 4-connected gr of 4-contractible edges aph, then . Theor of 4-contractible edges em C (Ando, E., Kawar of 4-contractible edges abayas hi & Kr of 4-contractible edges ies ell). If 4-contractible edges is a 4-connected gr of 4-contractible edges aph, then . Theor of 4-contractible edges em C (Ando, E., Kawar of 4-contractible edges abayas hi & Kr of 4-contractible edges ies ell). If 4-contractible edges is a 4-connected gr of 4-contractible edges aph, then . The f 4-contractible edges ollowing r of 4-contractible edges es ult has r of 4-contractible edges ecently been obtained. Theor of 4-contractible edges em D (Kotani & Nakamur of 4-contractible edges a). If 4-contractible edges is a 4-connected gr of 4-contractible edges aph, then . ver of 4-contractible edges tices

: In , r of 4-contractible edges eplace each ver of 4-contractible edges tex by , r of 4-contractible edges eplace each edge indicated by a br of 4-contractible edges oken line by In , r of 4-contractible edges eplace each ver of 4-contractible edges tex by , r of 4-contractible edges eplace each edge indicated by a br of 4-contractible edges oken line by

ver of 4-contractible edges tices : and r of 4-contractible edges eplace each edge indicated by a s olid line by 11 Let denote the r of 4-contractible edges es ulting gr of 4-contractible edges aph. 12

21 1 22 31 32 2 :1 : 1 =2(n+1)+4 =2n+6

Each has degr of 4-contractible edges ee .

: :31 : 1 Each has degr of 4-contractible edges ee . Each has degr of 4-contractible edges ee .

: :31 : 1 Each has degr of 4-contractible edges ee .

. Each Each Each Each has degr of 4-contractible edges ee . has degr of 4-contractible edges ee . has degr of 4-contractible edges ee . has degr of 4-contractible edges ee.

deg ( )=11 2+36+2 ( 2 +9 )=26 + 5 ( ) 1 1 | ( )|= 13 ( 26 n+78 )= 13 ( ( 5 ( )

deg ( ) +24 ) ) Conjectur of 4-contractible edges e. If 4-contractible edges is a 4-connected gr of 4-contractible edges aph f 4-contractible edges or of 4-contractible edges which is s uf 4-contractible edges f 4-contractible edges iciently lar of 4-contractible edges ge, then. Theor of 4-contractible edges em. If 4-contractible edges is a 4-connected gr of 4-contractible edges aph, then . Outline of 4-contractible edges the pr of 4-contractible edges oof 4-contractible edges . Set

Find a s ubs et of 4-contractible edges s uch that , and .

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