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arXiv · 2005.07224

On a generalized Erd\H{o}s-Rademacher problem

Xizhi Liu·Dhruv Mubayi
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Abstract

The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers $s,t$ and an $n$-vertex graph $G$ with $\lfloor n^2/4 \rfloor +t$ edges and triangle covering number $s$, we determine (for large $n$) sharp bounds on the minimum number of triangles in $G$ and also describe the extremal constructions. Similar results are proved for cliques of larger size and color critical graphs. This extends classical work of Rademacher, Erd\H os, and Lov\'asz-Simonovits whose results apply only to $s \le t$. Our results also address two conjectures of Xiao and Katona. We prove one of them and give a counterexample and prove a modified version of the other conjecture.

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Xizhi Liu, Dhruv Mubayi. 2020-05-14. On a generalized Erd\H{o}s-Rademacher problem. https://arxiv.org/abs/2005.07224

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