Triangles in Ks-saturated graphs with minimum degree t

Authors

Craig Timmons, California State University, SacramentoFollow
Benjamin Cole, California State University, SacramentoFollow
Albert Curry, California State University, SacramentoFollow
David Davini, University of California, Los AngelesFollow

Abstract

For n ≥ 15, we prove that the minimum number of triangles in an n-vertex K₄-saturated graph with minimum degree 4 is exactly 2n-4, and that there is a unique extremal graph. This is a triangle version of a result of Alon, Erdos, Holzman, and Krivelevich from 1996. Additionally, we show that for any s > r ≥ 3 and t ≥ 2 (s-2)+1, there is a K_s-saturated n-vertex graph with minimum degree t that has \binom{ s-2}{r-1}2^{r-1} n + c_{s,r,t} copies of K_r. This shows that unlike the number of edges, the number of K_r's (r >2) in a K_s-saturated graph is not forced to grow with the minimum degree, except for possibly in lower order terms.

Creative Commons License

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

Recommended Citation

Timmons, Craig; Cole, Benjamin; Curry, Albert; and Davini, David (2020) "Triangles in Ks-saturated graphs with minimum degree t," Theory and Applications of Graphs: Vol. 7: Iss. 1, Article 2.
DOI: 10.20429/tag.2020.070102
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol7/iss1/2