Year: 2024
Type: preprint
Abstract: For a group $Γ$, a $Γ$-labelled graph is an undirected graph $G$ where every orientation of an edge is assigned an element of $Γ$ so that opposite orientations of the same edge are assigned inverse elements. A path in $G$ is non-null if the product of the labels along the path is not the neutral element of $Γ$. We prove that for every finite group $Γ$, non-null $S$-$T$ paths in $Γ$-labelled graphs exhibit the half-integral Erdős-Pósa property. More precisely, there is a function $f$, depending on $Γ$, such that for every $Γ$-labelled graph $G$, subsets of vertices $S$ and $T$, and integer $k$, one of the following objects exists: a family $\cal F$ consisting of $k$ non-null $S$-$T$ paths in $G$ such that every vertex of $G$ participates in at most two paths of $\cal F$; or a set $X$ consisting of at most $f(k)$ vertices that meets every non-null $S$-$T$ path in $G$. This in particular proves that in undirected graphs $S$-$T$ paths of odd length have the half-integral Erdős-Pósa property.
Source: arXiv (Cornell University)
Institutions: Humboldt-Universität zu Berlin, Adam Mickiewicz University in Poznań, Lyon 1 Université, École Normale Supérieure de Lyon, Centre National de la Recherche Scientifique, Laboratoire de l'Informatique du Parallélisme, Institute for Basic Science, University of Warsaw, Max Planck Institute for Informatics, Université Libre de Bruxelles, Nicolaus Copernicus University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funders: European Commission, Deutscher Akademischer Austauschdienst, Deutsche Forschungsgemeinschaft, Agence Nationale de la Recherche, Bundesministerium für Bildung und Forschung, Fonds De La Recherche Scientifique - FNRS, Uniwersytet Warszawski, Institute for Basic Science, H2020 European Research Council
OpenAlex ID: W4402706563