Erde, Joshua; Gollin, J. Pascal; Joó, Attila Enlarging vertex-flames in countable digraphs. (English) Zbl 1473.05106 J. Comb. Theory, Ser. B 151, 263-281 (2021). Summary: A rooted digraph is a vertex-flame if for every vertex \(v\) there is a set of internally disjoint directed paths from the root to \(v\) whose set of terminal edges covers all ingoing edges of \(v\). It was shown by L. Lovász [ibid. 15, 174–177 (1973; Zbl 0264.05113)] that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. G. Calvillo-Vives [Optimum branching systems. Waterloo, ON: University of Waterloo (PhD Thesis) (1978)] rediscovered and extended this theorem proving that every vertex-flame of a given finite rooted digraph can be extended to be large. The analogue of Lovász’ result for countable digraphs was shown by the third author where the notion of largeness is interpreted in a structural way as in the infinite version of Menger’s theorem. We give a common generalisation of this and Calvillo-Vives’ result by showing that in every countable rooted digraph each vertex-flame can be extended to a large vertex-flame. Cited in 1 Document MSC: 05C20 Directed graphs (digraphs), tournaments 05C30 Enumeration in graph theory 05C63 Infinite graphs Keywords:infinite digraph; local connectivity; flame Citations:Zbl 0264.05113 PDFBibTeX XMLCite \textit{J. Erde} et al., J. Comb. Theory, Ser. B 151, 263--281 (2021; Zbl 1473.05106) Full Text: DOI arXiv References: [1] Aharoni, R.; Berger, E., Menger’s theorem for infinite graphs, Invent. Math., 176, 1, 1-62 (2009), MR2485879 · Zbl 1216.05092 [2] Calvillo-Vives, G., Optimum branching systems (1978), University of Waterloo, Ph.D. Thesis [3] Diestel, R., Graph Theory, Graduate Texts in Mathematics, vol. 173 (2017), Springer: Springer Berlin, MR3644391 · Zbl 1375.05002 [4] Joó, A., Packing countably many branchings with prescribed root-sets in infinite digraphs, J. Graph Theory, 87, 1, 96-107 (2018), MR3729839 · Zbl 1380.05162 [5] Joó, A., Vertex-flames in countable rooted digraphs preserving an Erdős-Menger separation for each vertex, Combinatorica, 39, 1317-1333 (2019) · Zbl 1449.05121 [6] Joó, A., The complete lattice of Erdős-Menger separations (2019), available at [7] Lovász, L., Connectivity in digraphs, J. Comb. Theory, Ser. B, 15, 174-177 (1973), MR325439 · Zbl 0264.05113 [8] Pym, J. S., The linking of sets in graphs, J. Lond. Math. Soc., 44, 542-550 (1969), MR234858 · Zbl 0167.52204 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.