Journal of Combinatorial Theory, Series B 83, 305319 (2001)
Almost All 3-Connected Graphs Contain a
Contractible Set of k Vertices
Matthias Kriesell
Inst. fu r Mathematik (A), Universita t Hannover, Welfengarten 1, D-30167 Hannover, Germany
Received December 22, 1999; published online August 17, 2001
McCuaig and Ota conjectured that every sufficiently large 3-connected graph G
contains a connected subgraph H on k vertices such that G&V(H) is 2-connected.
We prove the weaker statement that every sufficiently large 3-connected graph G
contains a not necessarily connected subgraph H on k vertices such that G&V(H)
is 2-connected.
2001 Elsevier Science
Key Words: connectivity; contractable subgraph; reduction; longest induced
path; longest cycle.
1. INTRODUCTION
All graphs considered here are supposed to be finite, undirected, and
loopless. To indicate the occurrence of multiple edges, we use the term mul-
tigraph, and to emphasize their absence, we use the attribute simple for a
graph. For terms not defined here we refer to [1, 3]. As a generalization
of Tutte's theorem stating that every 3-connected graph on at least 5 ver-
tices contains a contractible edge [6], McCuaig and Ota conjectured in
[5]
Conjecture 1 [5]. For every natural number k there exists a natural
number f (k) such that every 3-connected graph G on at least f (k) vertices
contains a connected subgraph on k vertices such that G&V(H) is 2-con-
nected.
For k=1, Conjecture 1 is trivial, for k=2 we have Tutte's Theorem [6].
From the results in [5] it follows f(3)=9, and from those in [4] we know
f(4)=8. Furthermore, the conjecture holds for several graph classes, such
as for maximal planar graphs, AT-free graphs, and 5-connected graphs of
bounded degree [4]. It is not yet decided for planar graphs or 4-connected
graphs. For odd k, the products C
k+1
_K
2
show f (k)>2k+2. For even k,
the graphs K
3, k
show f(k)>k+3, but I know only ``sporadic'' examples
showing that f (k) is substantially larger than k+3. The Petersen graph, for
example, establishes f(6)>10. More generally, (3, g)-cages, i.e. smallest
doi:10.1006Âjctb.2001.2060, available online at http:ÂÂwww.idealibrary.com on
305
0095-8956Â01 35.00
2001 Elsevier Science
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