Invalid Counterexample in Graph Theory

A Counterexample in Sturm-Liouville Completeness Theory

We give an example of an indefinite weight Sturm-Liouville problem whose eigenfunctions form a Riesz basis under Dirichlet boundary conditions but not under anti-periodic boundary conditions.

A counterexample to the pseudo 2-factor isomorphic graph conjecture

A graph G is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of G. Abreu et al. [1] conjectured that K3,3, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs (Abreu et al., Journal of Combinatorial Theory, Series B, 2008, Conjecture 3.6). Using a computer ...

Invalid Recipes

A Counterexample in the Theory of Local Zeta Functions

A counterexample to a result on the tree graph of a graph

Given a set of cycles C of a graph G, the tree graph of G, defined by C, is the graph T (G,C) whose vertices are the spanning trees of G and in which two trees R and S are adjacent if R∪S contains exactly one cycle and this cycle lies in C. Li et al. [Discrete Math. 271 (2003), 303–310] proved that if the graph T (G,C) is connected, then C cyclically spans the cycle space of G. Later, Yumei Hu ...