4 research outputs found
Lower bounds for on-line graph colorings
We propose two strategies for Presenter in on-line graph coloring games. The
first one constructs bipartite graphs and forces any on-line coloring algorithm
to use colors, where is the number of vertices in the
constructed graph. This is best possible up to an additive constant. The second
strategy constructs graphs that contain neither nor as a subgraph
and forces colors. The best known
on-line coloring algorithm for these graphs uses colors
Circumference, chromatic number and online coloring
ErdAs conjectured that if G is a triangle free graph of chromatic number at least ka parts per thousand yen3, then it contains an odd cycle of length at least k (2-o(1)) [13,15]. Nothing better than a linear bound ([3], Problem 5.1.55 in [16]) was so far known. We make progress on this conjecture by showing that G contains an odd cycle of length at least Omega(k log logk). ErdAs' conjecture is known to hold for graphs with girth at least five. We show that if a graph with girth four is C (5) free, then ErdAs' conjecture holds. When the number of vertices is not too large we can prove better bounds on chi. We also give bounds on the chromatic number of graphs with at most r cycles of length 1 mod k, or at most s cycles of length 2 mod k, or no cycles of length 3 mod k. Our techniques essentially consist of using a depth first search tree to decompose the graph into ordered paths, which are then fed to an online coloring algorithm. Using this technique we give simple proofs of some old results, and also obtain several other results. We also obtain a lower bound on the number of colors which an online coloring algorithm needs to use to color triangle free graphs