Monochromatic paths in edge-colored graphs

International Journal of Analysis and Applications, 2020

For a graph G = (V, E), a vertex coloring (or, simply, a coloring) of G is a function C : V (G) → {1, 2, ..., k} (using the non-negative integers {1, 2, ..., k} as colors). In this research work, we introduce a new type of graph coloring called monotone coloring, along with this new coloring, we define the monotone chromatic number of a graph and establish some related new graphs. Basic properties and exact values of the monotone chromatic number of some graph families, like standard graphs, Kragujevac trees and firefly graph are obtained. Also, we get a characterization for bipartite graphs by defining the monotone bipartite graph. Exact values of the monotone chromatic number for some special case of Cartesian product of graphs are found. Finally, upper and lower bounds for monotone chromatic number of the Cartesian product for non trivial connected graphs are presented.

Combinatorica, 2012

Extending Furstenberg's ergodic theoretic proof for Szemerédi's theorem on arithmetic progressions, proved the following qualitative result. For every d and k, there exists an integer N such that no matter how we color the vertices of a complete binary tree TN of depth N with k colors, we can find a monochromatic replica of T d in TN such that (1) all vertices at the same level in T d are mapped into vertices at the same level in TN ; (2) if a vertex x ∈ V (T d ) is mapped into a vertex y in TN , then the two children of x are mapped into descendants of the two children of y in TN , respectively; and (3) the levels occupied by this replica form an arithmetic progression in {0, 1, . . . , N − 1}. This result and its density versions imply van der Waerden's and Szemerédi's theorems, and laid the foundations of a new Ramsey theory for trees.

International Journal of Mathematics and Mathematical Sciences, 1987

Discrete Mathematics, 1982

Periodica Mathematica Hungarica, 1973

European Journal of Combinatorics, 2021

An ordered hypergraph is a hypergraph H with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph G in H must respect the specified order on V (G). In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The t-color on-line size Ramsey numberR t (G) of an ordered hypergraph G is the minimum number of edges Builder needs to play (on a large ordered set of vertices) to force Painter using t colors to produce a monochromatic copy of G. The monotone tight path P (k) r is the ordered hypergraph with r vertices whose edges are all sets of k consecutive vertices. We obtain good bounds onR t (P (k) r). Letting m = r − k + 1 (the number of edges in P (k) r), we prove m t−1 /(3 √ t) ≤R t (P (2) r) ≤ tm t+1. For general k, a trivial upper bound is R k , where R is the least number of vertices in a k-uniform (ordered) hypergraph whose t-colorings all contain P (k) r (and is a tower of height k − 2). We prove R/(k lg R) ≤R t (P (k) r) ≤ R(lg R) 2+ε , where ε is a positive constant and tm is sufficiently large in terms of ε −1. Our upper bounds improve prior results when t grows faster than m/ log m. We also generalize our results to ℓ-loose monotone paths, where each successive edge begins ℓ vertices after the previous edge.

Acta Mathematica Hungarica, 2020

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly connected subgraphs all of whose edges are colored by the same color.

Duke Mathematical Journal, 2013

Ramsey's theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique of order 1 2 log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of Rödl, we show that there is a constant c > 0 such that every 2-coloring of the edges of the complete graph on {2, 3, ..., n} contains a monochromatic clique S for which the sum of 1/ log i over all vertices i ∈ S is at least c log log log n. This is tight up to the constant factor c and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every k there is an n such that the following holds. For every permutation π of 1,. .. , k − 1, every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique a 1 <. .. < a k with a π(1)+1 − a π(1) > a π(2)+1 − a π(2) >. .. > a π(k−1)+1 − a π(k−1). That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.

Graphs and Combinatorics, 2019

While investigating odd-cycle free hypergraphs, Győri and Lemons introduced a colored version of the classical theorem of Erdős and Gallai on P k-free graphs. They proved that any graph G with a proper vertex coloring and no path of length 2k + 1 with end vertices of different colors has at most 2kn edges. We show that Erdős and Gallai's original sharp upper bound of kn holds for their problem as well. We also introduce a version of this problem for trees and present a generalization of the Erdős-Sós conjecture.

2010

Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.