Let H be a connected graph and G be a supergraph of H . It is trivial that for any k-flow (D, f ) of G, the restriction of (D, f ) on the edge subset E(G/H) is a k-flow of the contracted graph G/H . However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow (D′, f ′) of the contracted graph G/H , whether or not the supergraph G admits a k-flow (D,...

We exhibit a separable commutative C*-algebra A such that A⊗K is homotopy equivalent to zero, without Mn(A) being so for any n ≥ 1.

McCuaig and Ota proved that every 3-connected graph G on at least 9 vertices admits a contractible triple, i. e. a connected subgraph H on three vertices such that G − V (H) is 2-connected. Here we show that every 3-connected graph G on at least 9 vertices has more than |V (G)|/10 many contractible triples. If, moreover, G is cubic, then there are at least |V (G)|/3 many contractible triples, w...

Contractible transformations of graphs consist of contractible gluing and deleting of vertices and edges of graphs. They partition all graphs into the family of homotopy classes. Contractible transformations do not change the Euler characteristic and the homology groups of graphs. In this paper we describe the minimal representatives of some homotopy classes and find the formula for computing t...

To any graph G we can associate a simplicial complex ∆(G) whose simplices are the complete subgraphs of G, and thus we say that G is contractible whenever ∆(G) is so. We study the relationship between contractibility and K-nullity of G, where G is called K-null if some iterated clique graph of G is trivial. We show that there are contractible graphs which are not K-null, and that any graph whos...