Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

Abstract

For every $n\in\mathbb{N}$ and $k\geqslant2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geqslant3$, it is known that the complete graph can be replaced by a graph $G$ with $\delta(G)\geqslant(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show that the maximum possible value of $\varepsilon_3$ is $\frac16$. This disproves a conjecture of Gyárfas and Sárközy.

A note about monochromatic components in graphs of large minimum degree

“…Let G be a graph on n vertices. If δ(G) ≥ 5 6 n − 1, then mc 3 (G) ≥ n 2 . Moreover, for every n, there exists a graph G on n vertices with δ(G) = ⌈ 5 6 n⌉ − 2 such that mc 3 (G) < n 2 .…”

“…If δ(G) ≥ 5 6 n − 1, then mc 3 (G) ≥ n 2 . Moreover, for every n, there exists a graph G on n vertices with δ(G) = ⌈ 5 6 n⌉ − 2 such that mc 3 (G) < n 2 . Note that the 3-colorings of graphs with δ(G) = ⌈ 5 6 n⌉ − 2 given by Guggiari and Scott and Rahimi have largest monochromatic components of order just under n 2 .…”

Large Monochromatic Components in Almost Complete Graphs and Bipartite Graphs

“…It was also conjectured in [7] that γ(t) could be as big as t/(t+1) 2 . This was disproved for t = 2 by Guggiari and Scott [4] and by Rahimi [8], and more recently for general t by DeBiasio and Krueger [1]. The constructions of graphs in [1,4,8] are based on modified affine planes.…”

“…This was disproved for t = 2 by Guggiari and Scott [4] and by Rahimi [8], and more recently for general t by DeBiasio and Krueger [1]. The constructions of graphs in [1,4,8] are based on modified affine planes. They have minimum degree at least (1 − t−1 t(t+1) )n − 2 and a (t + 1)-edge coloring in which each monochromatic component is of order less than n/t.…”

“…They proved that γ(t, n 1 , n 2 ) (n 1 /n 2 ) 3 /(128t 5 ) suffices. For both Questions 1 and 2 the t = 2 case is solved completely in [4,8] and [2], respectively. They obtained γ(2) = 1/6, γ(1, n 1 , n 2 ) < 1/2, and γ(2, n 1 , n 2 ) < 1/3 (independently of n), and these constants are the best possible.…”