Gallai colorings of non-complete graphs

Gallai-colorings of complete graphs – edge colorings such that no triangle is colored with three distinct colors – occur in various contexts such as the theory of partially ordered sets (in Gallai’s original paper), information theory and the theory of perfect graphs. We extend here Gallai-colorings to non-complete graphs and study the analogue of a basic result – any Gallai-colored complete graph has a monochromatic spanning tree – in this more general setting. We show that edge colorings of a graph H without multicolored triangles contain monochromatic connected subgraphs with at least (α(H)2 + α(H) − 1)−1|V (H)| vertices, where α(H) is the independence number of H . In general, we show that if the edges of an r-uniform hypergraph H are colored so that there is no multicolored copy of a fixed F then there is a monochromatic connected subhypergraph H1 ⊆ H such that |V (H1)| ≥ c|V (H)|where c depends only on F , r , and α(H). © 2009 Elsevier B.V. All rights reserved. 1. Gallai-colorings of non-complete graphs Edge colorings of complete graphs in which no triangle is colored with three distinct colors were called Gallai-partitions in [10], Gallai-colorings in [7,6]. More than just the term, the concept occurs again and again in relation to deep structural properties of fundamental objects. An important result in Gallai’s original paper [4] – translated to English and endowed by comments in [11] – can be reformulated in terms of Gallai-colorings. Further occurrences are related to generalizations of the perfect graph theorem [2], or applications in information theory [9]. In this paperwe start investigatingwhether Gallai-colorings can be fruitfully extended from complete graphs to arbitrary graphs, i.e. we say that an edge coloring of a graph G is a Gallai-coloring – or G-coloring – if no triangle of G is colored with three distinct colors. In particular, every edge coloring of a triangle-free graph is a G-coloring. A less obvious example can be obtained by considering a labeling of the vertices of a graph of order n by 1, 2, . . . , n and for all 1 ≤ i < j ≤ n color the edge ij by color i. A basic remark of Erdős and Rado states that in any coloring of the edges of a complete graph with two colors there is a monochromatic spanning tree. This remains true for G-colorings of complete graphs as proved in [1], see also [6]. Our starting point is another generalization of the above remark, we state it as Theorem 1. Let α(H) be the independence number of H , that is, the maximum size of an independent set, set of vertices not containing both endpoints of any edge. Theorem 1. If the edges of an arbitrary graph H are colored with two colors, there exists a monochromatic subtree T ⊂ H with at least α(H)−1|V (H)| vertices. We derive Theorem 1 from König’s theorem and note that Theorem 1 can be extended for r-colorings as well, with (r − 1)α(H) in the role of α(H), this more general form can be obtained from Füredi’s result [3] on fractional transversals ∗ Corresponding author at: Computer Science Department, Worcester Polytechnic Institute, Worcester, MA, 01609, USA. E-mail addresses: [email protected] (A. Gyárfás), [email protected] (G.N. Sárközy). 0012-365X/$ – see front matter© 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2009.10.013 Author's personal copy 978 A. Gyárfás, G.N. Sárközy / Discrete Mathematics 310 (2010) 977–980 (see [5]). Notice that Theorem 1 is sharp if α(H) is a divisor of |V (H)|, and simply consider α(H) disjoint monochromatic complete graphs of equal order. Our main result is an analog of Theorem 1 for G-colorings. Theorem 2. If the edges of an arbitrary graph H are G-colored, there exists a monochromatic subtree with at least (α2(H) + α(H)− 1)−1|V (H)| vertices and a monochromatic double star with at least (α2(H)+ α(H)− 3 ) −1 |V (H)| vertices. In fact, the coefficient of |V (H)| in Theorem 2 is not very far from the truth, showing that the bound of Theorem 1 does not extend to G-colorings. Indeed, consider a triangle-free graph H = H(α) such that α(H) = α and has as many vertices as possible — in other words, H is a so called Ramsey-graph with R(3, α + 1) − 1 vertices. It is well-known that |V (H)| is ‘‘almost’’ quadratic in α, its order of magnitude is α 2 logα (see [8]). One can get a trivial G-coloring onH by coloring each edge of H with a different color. Then substituting into each vertex ofH a G-colored complete graph Kp, we get a G-colored graphH on p|V (H)| vertices, with α(H) = α and with largest monochromatic connected subgraph at most 2p-order of magnitude logα α2 |V (H)|vertices. The problem of determining f (α), the largest value such that every G-colored graph H has a monochromatic connected subgraph with at least f (α(H))|V (H)| vertices remains open even for α = 2. From Theorem 2 and from the construction above we get 1 (α2 + α − 1) ≤ f (α) ≤ c logα α2 where c is a constant, coming from Kim’s [8] estimate of R(3, α + 1). For α = 2 we have the example above from coloring the edges of a C5 with five distinct colors, substituting arbitrarily G-colored complete graphs of equal size to the vertices. However, this coloring is not the best for α = 2. We may take another Ramsey graph, H8, the complement of the Wagner graph, its missing edges form an eight cycle 1, 2, . . . , 8 plus its main diagonals. (The graph H8 is a smallest graph with α = 2, ω = 3, see for example [13].) The edges of H8 can be G-colored without a monochromatic connected subgraph on four vertices, by using color i on the edges (i, i+ 2), (i, i+ 5) for i = 1, 2, . . . , 8 (modulo 8 arithmetic). Using substitutions into this coloring of H8 and applying Theorem 2 we have Corollary 1. 5 ≤ f (2) ≤ 3 8 . Theorem 2 shows that colorings of G without multicolored K3 have monochromatic connected subgraphs with order proportional to V (G). This property remains true in a much more general setting. Extending the definition from graphs, let α(H) denote themaximumm such thatH containsm vertices that do not contain any edge ofH . A hypergraph is connected if both parts of every nontrivial 2-partition of its vertex set have nonempty intersection with some edge of the hypergraph. Theorem 3. Suppose that the edges of an r-uniform hypergraphH are colored so that H does not contain multicolored copies of an r-uniform hypergraph F . Then there is a monochromatic connected subhypergraphH1 ⊆ H such that |V (H1)| ≥ c|V (H)| where c depends only on F , r, and α(H) (thus does not depend onH).