Turán Graphs and the Number of Colorings

Edge-colorings of graphs avoiding fixed monochromatic subgraphs with linear Turán number

Edge colorings of graphs avoiding some fixed monochromatic subgraph with linear Turán number

Let F be a graph and k be a positive integer. With a graph G, we associate the quantity ck,F (G), the number of k-colorings of the edge set of G with no monochromatic copy of F . Consider the function ck,F : N −→ N given by ck,F (n) = max{ck,F (G) : |V (G)| = n}, the maximum of ck,F (G) over all graphs G on n vertices. In this paper we study the asymptotic behavior of ck,F and describe the extr...

Perfect $2$-colorings of the Platonic graphs

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

Hypergraphs and Geometry

For any xed graph F , we say that a graph G is F -free if it does not contain F as a subgraph. We denote by ex(n, F ) the maximum number of edges in a n-vertex graph which is F -free, known as the Turán number of F . In 1974, Erd®s and Rothschild considered a related question where we count the number of certain colorings. Given an integer r, by an r-coloring of a graph G we mean any r-edgecolo...

On splittable colorings of graphs and hypergraphs

The notion of a split coloring of a complete graph was introduced by Erdős and Gyárfás [7] as a generalization of split graphs. In this paper, we offer an alternate interpretation by comparing such a coloring to the classical Ramsey coloring problem via a two-round game played against an adversary. We show that the techniques used and bounds obtained on the extremal (r,m)-split coloring problem...