Large monochromatic components in 3-colored non-complete graphs

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Abstract

We show that in every 3-coloring of the edges of a graph G of order N such that δ(G)5N61, there is a monochromatic component of order at least N/2. We also show that this result is best possible.

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 KN there is a monochromatic component of order at least Nk1. Here we consider k=3, and extend this result to graphs of large minimum degree. Note that for k=2, Gyárfás and Sárközy [4] proved that in every 2-coloring of the edges of a graph G with N vertices and δ(G)3N/4, there is a monochromatic component with at least δ(G)+1 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 k=3 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 k3, if δ(G)(1k1k2)N, then in every k-coloring of the edges of G, there exists a monochromatic component of order at least Nk1. In [5], they showed that for a graph G of order N and with δ(G)9N/10, every 3-coloring of the edges of G contains a monochromatic component of order at least N/2. DeBiasio, Krueger and Sárközy [2] obtained the same result for a graph G with δ(G)7N/8 (see also [7]). We disprove this conjecture (for k=3) by showing that 5N/61 is the correct minimum degree threshold for three colors (and not 7N/9). Our goal is to show the following.

Theorem 1

Let G=(V,E) be a graph on N vertices. If δ(G)5N/61, then in every three coloring of the edges of G there exists a monochromatic component of order at least N/2.

We first show that in the case when N>6 and 6 divides N, there exists a graph G with |G|=N and δ(G)=5N/62 and a 3-coloring of the edges of G such that every monochromatic component has fewer than N/2 vertices.

Let {v1,...,v6} denote the vertices of complete graph K6. Let us remove v1v5, v2v4 and v3v6 from K6 to obtain the graph H. Color v1v2, v2v3, v1v3 and v4v6 with blue, v3v4, v4v5, v3v5 and v1v6 by red and v2v5, v2v6, v5v6 and v1v4 by green.

Now in the 3-colored graph H, replace v1, v4 and v6 by sets V1, V4 and V6 each consisting of N/6+1 vertices and v2, v3 and v5 by sets V2, V3 and V5 each consisting of N/61 vertices. All of the edges inside Vi's and all of the edges between Vi and Vj (if vivj is an edge of H) are present. We color the edges inside Vi's arbitrarily and the edges between Vi and Vj inherit the color of the vivj edge. We obtain a 3-coloring of the edges of a graph G with |G|=N and δ(G)=5N/62 such that its largest monochromatic component contains N/21 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 uV(G), N¯(u) 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 m,nN and c[0,1]. If G is a bipartite graph with part-sizes m and n, and |E(G)|cmn, then G has a component of order at least c(m+n).

Lemma 3

Let G=(V,E) be a bipartite graph with bipartition V=V1V2 where |V1||V2| and for some δ>0, let |V1|>δ and |V2|>3δ/2. If every vertex of each part has at most δ non-neighbors in the other part, then V(G) 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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