Abstract
M.Kriesell conjectured that there existed b, h such that every 5-connected graph G with at least b vertices can be contracted to a 5-connected graph G0 such that 0<VG−VG0<h. We show that this conjecture holds for vertex transitive 5-connected graphs.
Highlights
All graphs considered here are supposed to be simple, finite, and undirected graphs
Let G/H stand for the graph obtained from G by contracting every component of H to a single vertex and replacing each resulting double edges by a single edge
A subgraph H of G is said to be k-contractible if G/H is still k-connected
Summary
All graphs considered here are supposed to be simple, finite, and undirected graphs. For a connected graph G, a subset T ⊆ V(G) is called a smallest separator if |T| κ(G) and G − T has at least two components. We will show that Conjecture 1 is true for vertex transitive 5-connected graphs. Conjecture 1 holds for 5-connected graphs which contain a contractible edge. In order to show that Conjecture 1 holds for vertex transitive 5-connected graphs, we have to show that all vertex transitive contraction-critical 5-connected graphs have a small contractible subgraph. The key point of this paper is to characterize the local structure of a vertex transitive contraction-critical 5-connected graph and, to find the contractible subgraph of it. For convenience, a vertex transitive contraction-critical 5-connected graph will be called a TCC-5-connected graph. Let G be a 5-connected vertex transitive graph which is neither K6 nor icosahedron, and G can be contracted to a 5-connected G′ such that 0 < |V(G)| − |V(G′)| < 3.
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