Independent sets and colorings of $K_{t,t,t}$-free graphs
Abhishek Dhawan, Oliver Janzer, Abhishek Methuku

TL;DR
This paper proves a long-standing conjecture that $K_{t,t,t}$-free graphs with bounded maximum degree have chromatic number $O(rac{ ext{max degree}}{ ext{log max degree}})$, and establishes large independent sets in such graphs, advancing understanding of their structure.
Contribution
It confirms the conjecture for all 3-colorable graphs $F$, specifically $K_{t,t,t}$, and proves a strong form of a related independence number conjecture for these graphs.
Findings
Proves the conjecture for all $K_{t,t,t}$-free graphs with bounded degree.
Establishes large independent sets of size $(1 - o(1)) n rac{ ext{log } d}{d}$ in $K_{t,t,t}$-free graphs.
Introduces a new variant of the R"odl nibble method for constructing independent sets.
Abstract
Alon, Krivelevich, and Sudakov conjectured in 1999 that every -free graph of maximum degree at most has chromatic number . This was previously known only for almost bipartite graphs, that is, for subgraphs of (verified by Alon, Krivelevich, and Sudakov themselves), while most recent results were concerned with improving the leading constant factor in the case where is almost bipartite. We prove this conjecture for all -colorable graphs , i.e. subgraphs of , representing the first progress toward the conjecture since it was posed. A closely related conjecture of Ajtai, Erd\H{o}s, Koml\'os, and Szemer\'edi from 1981 asserts that for every graph , every -vertex -free graph of average degree contains an independent set of size . We prove this conjecture in a strong form for all…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Topological and Geometric Data Analysis
