Rainbow Tur\'an numbers for short brooms
John Byrne, E.G.K.M Gamlath, Anastasia Halfpap, Sydney Miyasaki, Alex, Parker

TL;DR
This paper investigates the maximum edges in large graphs that avoid rainbow copies of certain broom trees with a length-3 handle, providing bounds and constructions that depend on the divisibility of the number of edges.
Contribution
It establishes new bounds on rainbow Turán numbers for broom trees with a length-3 handle and shows their dependence on the divisibility properties of the number of edges.
Findings
Improved upper bounds on rainbow Turán numbers for broom trees with length-3 handles.
Construction of graphs asymptotically matching these bounds for specific cases.
Demonstration of the influence of divisibility properties of the number of edges on these bounds.
Abstract
A graph is rainbow--free if it admits a proper edge-coloring without a rainbow copy of . The rainbow Tur\'an number of , denoted , is the maximum number of edges in a rainbow--free graph on vertices. We determine bounds on the rainbow Tur\'an numbers of stars with a single edge subdivided twice; we call such a tree with total edges a -edge \textit{broom} with length- handle, denoted by . We improve the best known upper bounds on in all cases where . Moreover, in the case where is odd and in a few cases when , we provide constructions asymptotically achieving these upper bounds. Our results also demonstrate a dependence of on divisibility properties of .
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Taxonomy
TopicsComputability, Logic, AI Algorithms
