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 of F. The singular Ramsey number Rs(F) is the smallest positive integer n such that, for every m at least 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.

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

TAG_Vol6_1_1_Supp.pdf (78 kB)
Supplemental Reference List with DOIs