Large monochromatic components in 3-colored non-complete graphs
Introduction
In a coloring of the edges of a graph G with k colors, a monochromatic component is a maximal subgraph that is connected in one of the colors. Gyárfás [3] showed that in every k-coloring of the edges of the complete graph there is a monochromatic component of order at least . Here we consider , and extend this result to graphs of large minimum degree. Note that for , Gyárfás and Sárközy [4] proved that in every 2-coloring of the edges of a graph G with N vertices and , there is a monochromatic component with at least vertices (see also [1]). They also showed that this result is sharp and thus complete graphs are the only graphs having the property that in every 2-coloring of the edges there exists a monochromatic component covering all vertices. But the results obtained for state that in every 3-coloring of the edges of a non-complete graph G with appropriately large minimum degree, there is a monochromatic component which contains at least half of the vertices of G.
Gyárfás and Sárközy [5], conjectured that for any graph G with N vertices and for all , if , then in every k-coloring of the edges of G, there exists a monochromatic component of order at least . In [5], they showed that for a graph G of order N and with , every 3-coloring of the edges of G contains a monochromatic component of order at least . DeBiasio, Krueger and Sárközy [2] obtained the same result for a graph G with (see also [7]). We disprove this conjecture (for ) by showing that is the correct minimum degree threshold for three colors (and not ). Our goal is to show the following. Theorem 1 Let be a graph on N vertices. If , then in every three coloring of the edges of G there exists a monochromatic component of order at least .
We first show that in the case when and 6 divides N, there exists a graph G with and and a 3-coloring of the edges of G such that every monochromatic component has fewer than vertices.
Let denote the vertices of complete graph . Let us remove , and from to obtain the graph H. Color , , and with blue, , , and by red and , , and by green.
Now in the 3-colored graph H, replace , and by sets , and each consisting of vertices and , and by sets , and each consisting of vertices. All of the edges inside 's and all of the edges between and (if is an edge of H) are present. We color the edges inside 's arbitrarily and the edges between and inherit the color of the edge. We obtain a 3-coloring of the edges of a graph G with and such that its largest monochromatic component contains vertices.
Here a vertex which has no edges incident with one of the colors, will be considered a monochromatic trivial component in that color. For a vertex , denotes the set of non-neighbors of u.
Section snippets
Main result
In the proof of Theorem 1, we shall use the following Lemmas. We start with a Lemma of Liu et al. [6] (see also [8]). Lemma 2 ([6]) Let and . If G is a bipartite graph with part-sizes m and n, and , then G has a component of order at least . Lemma 3 Let be a bipartite graph with bipartition where and for some , let and . If every vertex of each part has at most non-neighbors in the other part, then can be covered by at most
Acknowledgement
I would like to thank the anonymous referees for their valuable comments and suggestions.
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