Kawarabayashi proved that for any integer k ≥ 4, every k-connected graph contains two triangles sharing an edge, or admits a k-contractible edge, or admits a k-contractible triangle. This implies Thomassen’s result that every triangle-free k-connected graph contains a k-contractible edge. In this paper, we extend Kawarabayashi’s technique and prove a more general result concerning k-contractibl...

One can define what it means for a compact manifold with corners to be a “contractible manifold with contractible faces.” Two combinatorially equivalent, contractible manifolds with contractible faces are diffeomorphic if and only if their 4-dimensional faces are diffeomorphic. It follows that two simple convex polytopes are combinatorially equivalent if and only if they are diffeomorphic as ma...

Let G be a locally compact group, 1 < p < ∞ and let ω be a weight function on G. Recently, we introduced the Lebesgue weighted L−algebra L ω (G). Here, we establish necessary and sufficient conditions for L ω (G) to be φ−contractible, pseudo-contractible or contractible. Moreover we give some similar results about L(G,ω).

We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite-dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras. Finally, we characterize Hermitian ∗-alg...

In this paper, we prove that every contraction-critical 2-connected infinite graph has no vertex of finite degree and contains uncountably many ends. Then, by investigating the distribution of contractible edges in a 2-connected locally finite infinite graph G, we show that the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of G is 2-arc-connec...

An edge in a k-connected graph G is called k-contractible if the graph G/e obtained from G by contracting e is k-connected. Generalizing earlier results on 3-contractible edges in spanning trees of 3-connected graphs, we prove that (except for the graphs Kk+1 if k ∈ {1, 2}) (a) every spanning tree of a k-connected triangle free graph has two k-contractible edges, (b) every spanning tree of a k-...