We say that a subgraph F of a graph G is singular if the degrees d G (v) are all equal or all distinct for the vertices v ∈ V (F). The singular Ramsey number Rs(F) is the smallest positive integer n such that, for every m ≥ n, in every edge 2-coloring of K m , at least one of the color classes contains F as a singular subgraph. In a similar flavor, the singular Turán number Ts(n, F) is defined as the maximum number of edges in a graph of order n, which does not contain F as a singular subgraph. In this paper we initiate the study of these extremal problems. We develop methods to estimate Rs(F) and Ts(n, F), present tight asymptotic bounds and exact results.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
- Numerical Analysis