Online Dominating Set

This paper is devoted to the online dominating set problem and its variants on trees, bipartite, bounded-degree, planar, and general graphs, distinguishing between connected and not necessarily connected graphs. We believe this paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against the other. In most cases, we obtain tight bounds on the competitive ratios. Our results show that requiring the graphs to be presented in a connected fashion allows the online algorithms to obtain provably better solutions. Furthermore, we get detailed information regarding the significance of the necessary requirement that online algorithms be incremental. In some cases, having to be incremental fully accounts for the online algorithm\u27s disadvantage

Grunbaum colorings of even triangulations on surfaces

A Grunbaum coloring of a triangulation G is a map c : E(G){1,2,3} such that for each face f of G, the three edges of the boundary walk of f are colored by three distinct colors. By Four Color Theorem, it is known that every triangulation on the sphere has a Grunbaum coloring. So, in this article, we investigate the question whether each even (i.e.,Eulerian) triangulation on a surface with representativity at least r has a Grunbaum coloring. We prove that, regardless of the representativity, every even triangulation on a surface F has a Grunbaum coloring as long as F is the projective plane, the torus, or the Klein bottle, and we observe that the same holds for any surface with sufficiently large representativity. On the other hand, we construct even triangulations with no Grunbaum coloring and representativity r=1,2, and 3 for all but finitely many surfaces. In dual terms, our results imply that no snark admits an even map on the projective plane, the torus, or the Klein bottle, and that all but finitely many surfaces admit an even map of a snark with representativity at least3