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 Kn there is a monochromatic connected component of order at least n 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 δ(G) > 3n 4 ensures a monochromatic connected component with at least δ(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−εk)n can replace complete graphs and still there is a monochromatic connected component of order at least n k−1 , in fact δ(G) > ( 1− 1 1000(k − 1)9 ) n suffices. Our second result is an improvement of this bound for k = 3. If the edges of G with δ(G) > 9n 10 are 3-colored, then there is a monochromatic component of order at least n2 . We conjecture that this can be improved to 7n 9 and for general k we conjecture the following: if k > 3 and G is a graph of order n such that δ(G) > ( 1− k−1 k2 ) n, then in any k-coloring of the edges of G there is a monochromatic connected component of order at least n k−1 .