An on-line graph coloring algorithm with sublinear performance ratio
Discrete Mathematics
Graphs with k odd cycle lengths
Discrete Mathematics
Lower bounds for on-line graph coloring
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Journal of Combinatorial Theory Series B
The number of cycle lengths in graphs of given minimum degree and girth
Discrete Mathematics
Triangle-free graphs with large chromatic numbers
Discrete Mathematics
Introduction to algorithms
Developments from a June 1996 seminar on Online algorithms: the state of the art
A bound on the chromatic number using the longest odd cycle length
Journal of Graph Theory
Cycle lengths in sparse graphs
Combinatorica
Cycle length parities and the chromatic number
Journal of Graph Theory
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Erd驴s conjectured that if G is a triangle free graph of chromatic number at least k驴3, then it contains an odd cycle of length at least k 2驴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 Ω(k log logk). Erd驴s' 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 Erd驴s' conjecture holds. When the number of vertices is not too large we can prove better bounds on 驴. 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.