Circumference, Chromatic Number and Online Coloring

TL;DR

This paper advances understanding of the relationship between chromatic number and odd cycle length in graphs, providing new bounds and techniques for online coloring, especially in triangle-free and girth-restricted graphs.

Contribution

It proves that graphs with high chromatic number contain long odd cycles of at least O(k log log k), and introduces a novel online coloring approach using depth-first search decompositions.

Findings

Graphs with high chromatic number contain long odd cycles.

New bounds on chromatic number for graphs with limited cycle structures.

Lower bounds on online coloring complexity for triangle-free graphs.

Abstract

Erd\"os conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound (\cite{gyarfas}, Problem 5.1.55 in \cite{West}) was so far known. We make progress on this conjecture by showing that $G$ contains an odd cycle of length at least $O(k\log\log k)$. Erd\"os' conjecture is known to hold for graphs with girth at least 5. We show that if a girth 4 graph is $C_5$ free, then Erd\"os' 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 $1\bmod k$, or at most $s$ cycles of length $2\bmod k$, or no cycles of length $3\bmod k$. 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 simpler results. We also obtain a lower bound on the number of colors an online coloring algorithm needs to use on triangle free graphs.