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# Florian Gut and Attila Joó, Large vertex-flames in uncountable digraphs

## Wednesday, June 30, 2021 @ 5:00 PM - 6:00 PM KST

Zoom ID: 869 4632 6610 (ibsdimag)

### Speaker

Florian Gut
Department of Mathematics, University of Hamburg, Germany
Attila Joó
Department of Mathematics, University of Hamburg, Germany
https://www.math.uni-hamburg.de/home/joo/

The local connectivity  $\kappa_D(r,v)$ from $r$ to $v$ is defined to be the maximal number of internally disjoint $r\rightarrow v$ paths in $D$. A spanning subdigraph $L$ of $D$ with $\kappa_L(r,v)=\kappa_D(r,v)$ for every $v\in V-r$ must have at least $\sum_{v\in V-r}\kappa_D(r,v)$ edges. It was shown by Lovász that, maybe surprisingly, this lower bound is sharp for every finite digraph. The optimality of an $L$ can be captured by the following characterization: For every $v\in V-r$ there is a system $\mathcal{P}_v$ of internally disjoint $r\rightarrow v$ paths in $L$ covering all the ingoing edges of $v$ in $L$ such that one can choose from  each $P\in \mathcal{P}_v$ either an edge or an internal vertex in such a way that the resulting set meets every $r\rightarrow v$ path of $D$. We prove that every digraph of size at most $\aleph_1$  admits such a spanning subdigraph $L$. The question if this remains true for larger digraphs remains open.

## Details

Date:
Wednesday, June 30, 2021
Time:
5:00 PM - 6:00 PM KST
Event Category:
Event Tags:
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## Venue

Zoom ID: 869 4632 6610 (ibsdimag)

## Organizer

O-joung Kwon (권오정)
기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209