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Remark on a paper of J. Zaks
Discrete Mathematics 26 (1979) MS 297-301. @ North-Holland Publishing Campany
COMMUNICATION
EMARK ON A PAPER OF 3. ZAKS Hansjoachim
WALTHER
Department of Mathematics, Democratic Republic
Technische Hdschule
Ilmenau, DDR -63 Ilmenau, German
Communicated by B. Griinbaum Final form received 16 February 1979 Let be YY,i = 0, 1,2, the family of all 3-valent 3-connected planar graphs having elementary k-gons only if k = i (mod 3). Improving a result of J. Zaks we can show the inequality log 22
o(~i)~a(so)S-
i = 1,2,
log 23 ’
for the Shortness Exponent UC@) of the family gi.
Definitions The Shortness (seeKI)
Exponent
a(%) : = lim inf GE%
a(%) of a family % of graphs has been defined by
log h(G) log
u(G)
’
where v(G) denotes the number of vertices of the graph G and h(G) denotes the length of a maximum circuit of G. @, i = 0, 1,2 denotes the family of all 3-valent 3-c( jnnected planar graphs (3-regular polyhedral graphs) having elementary k gons only if k = i (mod 3). Problems and results J. Zaks [2] has shown lag 89 Q(@) = a(S2)
l
l
l
2
and B. CMnbaum
(private communication
to J. Zaks) has shown
o(%O) < 1. In this paper we will prove the following theorem. 297
H. Walther
298
(b)
(a) Fig. 1.
0(~‘)S~(~“)S-=
log 22 log 23
0.985823..
.,
i =
1,2.
Proof. (1) We will prove a(%“) slog 22/leg 23 (see [3]). Let us consider the fragment L of Fig. l(a). It is easy to show that the following property holds. I’hcre is no path through L, connecting any two half-edges and c,Jntaining all 23 black vertices. JILw vt construct a suitable sequence {Gi) of graphs in go, as follows. ti) Cr(:is the tetraedergraph having only one black; and three white vertices (see Fig. 1(‘h)). [ii) Gz,., is obtained from Gz by replacing every black vertex by a copy of the creation L from Fig. l(a). It is obvious that there are only k-gons with k s 0 (mod 3) in Gz, furthermore Gz is 3-valent and a-connected. A simple estimation similar to the arguments in [l, 2) yields log MG!!) __log 22 log 23’ a- log u(G:) lim
12;
r/N we will prove ~(%‘)SO(~~)
Remark on a paper of J. Zaks
299
n
4
0 0
0
v
H. WaIther
Fig. 3. (i)
Let be (GE} a sequence
of graphs in C!?O with
(ii) By replacing some conveniently choosen vertices of GE by suitable fragments (Fig. 2.) we get a sequence (H,!] s 3’. (iii) For realization of these replacements we consider a map
from the set F of p-gons of G’,’into the set 24!’of vertices. We only require that the vertex q(F) is incident with the face F. Let Y be an arbitrary vertex in GE. The following four cases Co, . . . , C, are possible. Ci : Thers arc cxaci”ly i faces Fl, . . . , Fi with the property q( F,) = Y, j = 1, . . . , i (i = 0 nxans that there is no face F with
remains unchanged, by a copy of Fig 2(a, b) according to Fig. 3, by 8 copy of Fig, a@, d) aamdhg to Fig, 4, by a copy of F’ig, 2(e,f),
The symbol in Fig, 2(b) (and analogously in 2(d) and 2(f)) means that the ink&r P; the frtigment contnina only k=g~n~ with k = 1 (mod 3) and the number
Remark on a paper
ofJ.
Zaks
301
of vertices between the three halfedges along the outer regions is = 1 (mod 3), 4 (mod 3). =2 lmod 3), resp. (vj T*hevertices are replaced in such a way that the resulting graph H,!, belongs to (8’. (vi) An easy estimation yields &(G~)Q.$H~)
and
h(HA)~25h(Gi).
hence it follows log h(Zfi) <‘E+*ogh(G~)
References [l] B. Griinbaum und H. Walther, Shortness exponents of families of graphs, J. Combinatorial Theory A, 14(1973) 364-385. [ 2] J. Zaks, Non-Hamiltonian non-Grinbergian graphs, Discrete Mathematics 17(197?) 3 17-32 1. [3] H. Walther und M. Schiiuble, ober den Shortness Exponenten normaler LardkPr!cln. “Beitrgge zur Graphentheorie und deren Anwendungen”, Oberhof 1977.
COMMUNICATION
EMARK ON A PAPER OF 3. ZAKS Hansjoachim
WALTHER
Department of Mathematics, Democratic Republic
Technische Hdschule
Ilmenau, DDR -63 Ilmenau, German
Communicated by B. Griinbaum Final form received 16 February 1979 Let be YY,i = 0, 1,2, the family of all 3-valent 3-connected planar graphs having elementary k-gons only if k = i (mod 3). Improving a result of J. Zaks we can show the inequality log 22
o(~i)~a(so)S-
i = 1,2,
log 23 ’
for the Shortness Exponent UC@) of the family gi.
Definitions The Shortness (seeKI)
Exponent
a(%) : = lim inf GE%
a(%) of a family % of graphs has been defined by
log h(G) log
u(G)
’
where v(G) denotes the number of vertices of the graph G and h(G) denotes the length of a maximum circuit of G. @, i = 0, 1,2 denotes the family of all 3-valent 3-c( jnnected planar graphs (3-regular polyhedral graphs) having elementary k gons only if k = i (mod 3). Problems and results J. Zaks [2] has shown lag 89 Q(@) = a(S2)
l
l
l
2
and B. CMnbaum
(private communication
to J. Zaks) has shown
o(%O) < 1. In this paper we will prove the following theorem. 297
H. Walther
298
(b)
(a) Fig. 1.
0(~‘)S~(~“)S-=
log 22 log 23
0.985823..
.,
i =
1,2.
Proof. (1) We will prove a(%“) slog 22/leg 23 (see [3]). Let us consider the fragment L of Fig. l(a). It is easy to show that the following property holds. I’hcre is no path through L, connecting any two half-edges and c,Jntaining all 23 black vertices. JILw vt construct a suitable sequence {Gi) of graphs in go, as follows. ti) Cr(:is the tetraedergraph having only one black; and three white vertices (see Fig. 1(‘h)). [ii) Gz,., is obtained from Gz by replacing every black vertex by a copy of the creation L from Fig. l(a). It is obvious that there are only k-gons with k s 0 (mod 3) in Gz, furthermore Gz is 3-valent and a-connected. A simple estimation similar to the arguments in [l, 2) yields log MG!!) __log 22 log 23’ a- log u(G:) lim
12;
r/N we will prove ~(%‘)SO(~~)
Remark on a paper of J. Zaks
299
n
4
0 0
0
v
H. WaIther
Fig. 3. (i)
Let be (GE} a sequence
of graphs in C!?O with
(ii) By replacing some conveniently choosen vertices of GE by suitable fragments (Fig. 2.) we get a sequence (H,!] s 3’. (iii) For realization of these replacements we consider a map
from the set F of p-gons of G’,’into the set 24!’of vertices. We only require that the vertex q(F) is incident with the face F. Let Y be an arbitrary vertex in GE. The following four cases Co, . . . , C, are possible. Ci : Thers arc cxaci”ly i faces Fl, . . . , Fi with the property q( F,) = Y, j = 1, . . . , i (i = 0 nxans that there is no face F with
remains unchanged, by a copy of Fig 2(a, b) according to Fig. 3, by 8 copy of Fig, a@, d) aamdhg to Fig, 4, by a copy of F’ig, 2(e,f),
The symbol in Fig, 2(b) (and analogously in 2(d) and 2(f)) means that the ink&r P; the frtigment contnina only k=g~n~ with k = 1 (mod 3) and the number
Remark on a paper
ofJ.
Zaks
301
of vertices between the three halfedges along the outer regions is = 1 (mod 3), 4 (mod 3). =2 lmod 3), resp. (vj T*hevertices are replaced in such a way that the resulting graph H,!, belongs to (8’. (vi) An easy estimation yields &(G~)Q.$H~)
and
h(HA)~25h(Gi).
hence it follows log h(Zfi) <‘E+*ogh(G~)
References [l] B. Griinbaum und H. Walther, Shortness exponents of families of graphs, J. Combinatorial Theory A, 14(1973) 364-385. [ 2] J. Zaks, Non-Hamiltonian non-Grinbergian graphs, Discrete Mathematics 17(197?) 3 17-32 1. [3] H. Walther und M. Schiiuble, ober den Shortness Exponenten normaler LardkPr!cln. “Beitrgge zur Graphentheorie und deren Anwendungen”, Oberhof 1977.