Threshold Functions for Asymmetric Ramsey Properties Involving Cliques

• Martin Marciniszyn
• Jozef Skokan
• Reto Spöhel
• Angelika Steger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4110)

Abstract

Consider the following problem: For given graphs G and F 1,..., F k , find a coloring of the edges of G with k colors such that G does not contain F i in color i. For example, if every F i is the path P 3 on 3 vertices, then we are looking for a proper k-edge-coloring of G, i.e., a coloring of the edges of G with no pair of edges of the same color incident to the same vertex.

Rödl and Ruciński studied this problem for the random graph G $$_{n,{\it p}}$$ in the symmetric case when k is fixed and F 1=...=F k =F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that pbn  − β for some constants b=b(F,k) and β= β(F). Their proof was, however, non-constructive. This result is essentially best possible because for pBn  − β, where B=B(F, k) is a large constant, such an edge-coloring does not exist. For this reason we refer to n  − β as a threshold function.

In this paper we address the case when F 1,..., F k are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k-edge-coloring of G n,p with pbn  − β for some constants b=b(F 1,..., F k , k) and β = β(F 1,..., F k ). Kohayakawa and Kreuter conjectured that $$n^{-\beta(F_1,\dots, F_k)}$$ is a threshold function in this case. This algorithm can be also adjusted to produce a valid k-coloring in the symmetric case.

Keywords

Random Graph Isomorphism Class Symmetric Case Threshold Function Isomorphic Graph

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Authors and Affiliations

• Martin Marciniszyn
• 1
• Jozef Skokan
• 2
• Reto Spöhel
• 1
• Angelika Steger
• 1
1. 1.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland
2. 2.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil