arXiv subjects
Publications and source records attributed to Mikhail S. Terekhov.
We prove the following ``symmetric analogue'' of Lovász's estimate (1975): if an $r$-partite hypergraph of rank $r\geqslant2$ has a cover of cardinality $n<\infty$, then it admits a cover of cardinality at most $nr/2$, which is invariant with respect to all automorphisms preserving the parts. We obtain also symmetric analogues of generalisations of Lovász's estimate due to Aharoni, Holzman, and Krivelevich (1996).
It is known that, if removing some $n$ edges from a graph $Γ$ destroys all subgraphs isomorphic to a given finite graph $K$, then all subgraphs isomorphic to $K$ can be destroyed by removing at most $|E(K)|\cdot n$ edges, which form a set invariant with respect to all automorphisms of $Γ$. We construct the first examples of (connected) graphs $K$ for which this estimate is not sharp. Our arguments are based on a ``weighted analogue'' of an earlier known estimate for the cost of symmetry.