No abstract

Let G be a graph and for any natural number r, s (G, r) denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, s (T , 1) (T ) + 1. Here we generalize this result and show that s (T , 2) (T ) + 1. Moreover, we show that if for any two vertices u and v with maximum degree d(u, v) 3, then s (T , 2) = (T ). Also for any tree T with (T ) 3 we prove that s (T , 3) 2 (T ) − 1. Finally, it is shown that for any graph G with no isolated edges, s (G, 1) 3 (G).

No abstract

We consider two-stage adjustable robust linear optimization problems with uncertain right hand side b belonging to a convex and compact uncertainty set U. We provide an a priori approximation bound on the ratio of the optimal affine (in b) solution to the optimal adjustable solution that depends on two fundamental geometric properties of U: (a) the "symmetry" and (b) the "simplex dilation factor" of the uncertainty set U and provides deeper insight on the power of affine policies for this class of problems. The bound improves upon a priori bounds obtained for robust and affine policies proposed in the literature. We also find that the proposed a priori bound is quite close to a posteriori bounds computed in specific instances of an inventory control problem, illustrating that the proposed bound is informative.

We show that the Eulerian-Catalan numbers enumerate Dyck permutations. We provide two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analogue of the Chung-Feller theorem.