The Covering Threshold of a Directed Acyclic Graph by Directed Acyclic Subgraphs
نویسندگان
چکیده
Let $H$ be a directed acyclic graph (dag) that is not rooted star. It known there are constants $c=c(H)$ and $C=C(H)$ such the following holds for $D_n$, complete on $n$ vertices. There set of at most $C\log n$ subgraphs $D_n$ covers every $H$-copy while $c\log does cover all $H$-copies. Here this dichotomy considerably strengthened.
 ${\vec G}(n,p)$ denote probability space graphs with vertices edge $p$. The fractional arboricity $a(H) = max \{\frac{|E(H')|}{|V(H')|-1}\}$, where maximum over non-singleton $H$. If \frac{|E(H)|}{|V(H)|-1}$ then totally balanced. Complete graphs, multipartite cycles, trees, and, in fact, almost proven that:
 
 dag $h$ $m$ edges which For $a^* > a(H)$ exists $c^* c^*(a^*,H) 0$ a.a.s. $G \sim {\vec G}(n,n^{-1/a^*})$ has property $X$ $c^*\log $G$ $H$-copies $G$. Moreover, $s(H) m/2 + O(m^{4/5}h^{1/5})$ stronger assertion any $X$: an no more than $s(H)$ its covered by each element $X$.
 balanced $0 < a^* a(H)$, single subgraph $H$-copies.
 As first result, note if $h=o(m)$ $s(H)=(1+o_m(1))m/2$ about half In infinitely many it $s(H)=m/2$, optimally. second requirement cannot, generally, relaxed.
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ژورنال
عنوان ژورنال: Electronic Journal of Combinatorics
سال: 2022
ISSN: ['1077-8926', '1097-1440']
DOI: https://doi.org/10.37236/11275