Let $D$ be a digraph, let $p \geq 1$ an integer, and $f: V(D) \to \mathbb{N}_0^p$ vector function with $f=(f_1,f_2,\ldots,f_p)$. We say that has $f$-partition if there is partition $(V_1,V_2,\ldots,V_p)$ of the vertex set such that, for all $i \in [1,p]$, digraph $D_i=D[V_i]$ weakly $f_i$-degenerate, is, in every nonempty subdigraph $D'$ $D_i$ $v$ $\min\{d_{D'}^+(v), d_{D'}^-(v)\} < f_i(v)$. In...

Let f be a nonnegative integer valued function on the vertex set of graph. A graph is strictly f-degenerate if each nonempty subgraph ? has v such that deg?(v)<f(v). In this paper, we define new concept, transversal, which generalizes list coloring, signed DP-coloring, L-forested-coloring, and (f1,f2,…,fs)-partition. cover G H with V(H)=?v?V(G)Xv, where Xv={(v,1),(v,2),…,(v,s)}; edge M=?uv?E(G)...

This paper shows the relationship between degeneracy degrees and multiplicities in linear programming models. The usual definition of degeneracy is restricted to vertices of a polyhedron. We introduce degeneracy for nonempty subsets of polyhedra and show that for linear programming models for which the feasible region contains at least one vertex holds that the dimension of the optimal face is ...

Quantum field theory in the near-horizon region of a black hole predicts the existence of an infinite number of degenerate modes. Such a degeneracy is regulated in the brick wall model by the introduction of a short distance cutoff. In this Letter we show that states of the brick wall model with non zero energy admit a further degeneracy for any given finite value of the cutoff. The black hole ...

It has long been thought that degeneracy in unfolding only concerned non-metric unfolding. Recently, Busing, Groenen, and Heiser have established that degeneracy occurs for all transformations that include estimation of an intercept and a slope. Consequently, degeneracy also plagues metric unfolding, since one member of the metric transformation family, the interval transformation, includes est...