Department of

# Mathematics

Seminar Calendar
for events the day of Friday, June 14, 2019.

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30


Friday, June 14, 2019

2:00 pm in 243 Altgeld Hall,Friday, June 14, 2019

#### Decomposing graphs into edges and triangles

###### Bernard Lidicky (Iowa State Math)

Abstract: We prove the following 30-year old conjecture of Gyori and Tuza: the edges of every $n$-vertex graph $G$ can be decomposed into complete graphs $C_1,\ldots,C_\ell$ of orders two and three such that $|C_1|+\cdots+|C_\ell|\le (1/2+o(1))n^2$. This result implies the asymptotic version of the old result of Erdos, Goodman and Posa that asserts the existence of such a decomposition with $\ell\le n^2/4$. We also discuss removing $o(1)$ term sharpening the result and possible extensions. The talk is based on joint works with Blumenthal, Kral, Martins, Pehova, Pikhurko, Pfender, Vole.