2024/03/14 by Li, Ruonan, Lu, Ruhui, Su, Xueli +1 · 1 citation
#05C05 #05C20 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2403.09082
An edge-colored graph G is called properly colored if every two adjacent edges are assigned different colors. A monochromatic triangle is a cycle of length 3 with all the edges having the same color. Given a tree T0, let T(n,T0) be the collection of n-vertex trees that are subdivisions of T0. It is conjectured that for each fixed tree T0, there is a function f(T0) such that for each integer n≥ f(T0) and each T∈ T(n,T0), every edge-colored complete graph Kn without containing monochromatic triangle must contain a properly colored copy of T. We confirm the conjecture in the case that T0 is a star. A weaker version of the above conjecture is also obtained. Moreover, to get a nice quantitative estimation of f(T0) when T0 is a star requires determining the constraint Ramsey number of a monochromatic triangle and a rainbow star, which is of independent interest.