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.
Highlights
Erdos and Rado noted that, for any graph G, either G or its complement is connected
Let G be a graph of order n with δ(G)
If the edges of G are k-coloured, there exists a monochromatic component of order at least n k−1
Summary
Erdos and Rado noted that, for any graph G, either G or its complement is connected. This is equivalent to the statement that every 2-edge-colouring of a complete graph contains a monochromatic spanning tree. Gyarfas [4] extended this result to k 3 colours In every k-edge-colouring of the complete graph on n vertices, there exists a monochromatic component of order. For k 3, the situation is different In this case, it is possible to remove some edges from a complete graph and still obtain a monochromatic component of order n k−1 in every k-edge-colouring. In every 3-colouring of the edges of G, there exists a monochromatic component of order the electronic journal of combinatorics 28(1) (2021), #P1.10. −2 and a 3-colouring of the edges of G such that every monochromatic component has order strictly less than This proves that Conjecture 3 is false for every n ∈ N and k = 3.
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