Theory and Applications of Graphs

Article Title

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 $n \geq 15$, we prove that the minimum number of triangles in an $n$-vertex

$K_4$-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, Erd\H{o}s, Holzman, and Krivelevich from 1996.

Additionally, we show that for any $s > r \geq 3$ and $t \geq 2 (s-2)+1$, 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