Large Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition

It is well-known that in every $k$-coloring of the edges of the complete graph $K_n$ there is a monochromatic connected component of order at least ${n\over k-1}$. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For $k=2$ the authors proved that $\delta(G)\ge{3n\over 4}$ ensures a monochromatic connected component with at least $\delta(G)+1$ vertices in every $2$-coloring of the edges of a graph $G$ with $n$ vertices. This result is sharp, thus for $k=2$ we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is  that for larger values of $k$ the situation is different, graphs of minimum degree $(1-\epsilon_k)n$ can replace complete graphs and still there is a monochromatic connected component of order at least ${n\over k-1}$, in fact $$\delta(G)\ge \left(1 - \frac{1}{1000(k-1)^9}\right)n$$ suffices. Our second result is an improvement of this bound for $k=3$. If the edges of $G$ with  $\delta(G)\geq {9n\over 10}$ are $3$-colored, then there is a monochromatic component of order at least ${n\over 2}$. We conjecture that this can be improved to ${7n\over 9}$ and for general $k$ we conjecture the following: if $k\geq 3$ and  $G$ is a graph of order $n$ such that $\delta(G)\geq \left( 1 - \frac{k-1}{k^2}\right)n$, then in any $k$-coloring of the edges of $G$ there is a monochromatic connected component of order at least ${n\over k-1}$.