Coloring locally sparse graphs
James Anderson, Abhishek Dhawan, Aiya Kuchukova

TL;DR
This paper generalizes local sparsity concepts in graphs to include subgraphs isomorphic to a fixed graph F, providing new bounds on chromatic number using the Rödl nibble method, applicable to list and correspondence coloring.
Contribution
It introduces the notion of (k, F)-local sparsity and proves a generalized chromatic bound for such graphs, extending previous results to broader classes of graphs and coloring models.
Findings
Generalized chromatic bound for (k, F)-locally-sparse graphs
Improved bounds for graphs free of certain bipartite subgraphs
Applicable to list and correspondence coloring settings
Abstract
A graph is -locally sparse if for each vertex , the subgraph induced by its neighborhood contains at most edges. Alon, Krivelevich, and Sudakov showed that for if a graph of maximum degree is -locally-sparse, then . We introduce a more general notion of local sparsity by defining graphs to be -locally-sparse for some graph if for each vertex the subgraph induced by the neighborhood of contains at most copies of . Employing the R\"{o}dl nibble method, we prove the following generalization of the above result: for every bipartite graph , if is -locally-sparse, then . This improves upon results of Davies, Kang, Pirot, and Sereni who consider the case when is a path.…
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Taxonomy
TopicsGraph Labeling and Dimension Problems
