Large Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition
Abstract
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 nk−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)≥3n4 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 nk−1, in fact δ(G)≥(1−11000(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)≥9n10 are 3-colored, then there is a monochromatic component of order at least n2. We conjecture that this can be improved to 7n9 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−1k2)n, then in any k-coloring of the edges of G there is a monochromatic connected component of order at least nk−1.