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Packing Trees with Constraints on the Leaf Degree

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Abstract

If m is a positive integer then we call a tree on at least 2 vertices an m-tree if no vertex is adjacent to more than m leaves. Kaneko proved that a connected, undirected graph G = (V, E) has a spanning m-tree if and only if for every \(X \subseteq V\) the number of isolated vertices of G − X is at most \(m|X|+{\rm max}\{0,|X| - 1\}\)—unless we have the exceptional case of \(G \simeq K_3\) and m = 1. As an attempt to integrate this result into the theory of graph packings, in this paper we consider the problem of packing a graph with m-trees. We use an approach different from that of Kaneko, and we deduce Gallai–Edmonds and Berge–Tutte type theorems and a matroidal result for the m-tree packing problem.

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  1. MTA-ELTE Egerváry Research Group (EGRES), Institute of Mathematics, Eötvös University, Budapest, Pázmány P.s.1/C, Hungary, H-1117

    Jácint Szabó

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  1. Jácint Szabó
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Correspondence to Jácint Szabó.

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Jácint Szabó: Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No.504438.

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Szabó, J. Packing Trees with Constraints on the Leaf Degree. Graphs and Combinatorics 24, 485–494 (2008). https://doi.org/10.1007/s00373-008-0804-x

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  • DOI: https://doi.org/10.1007/s00373-008-0804-x

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