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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 an 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|+(|X|-1)+ - unless we are at the exceptional case of G= K3 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 to that of Kaneko, and we deduce Gallai--Edmonds and Berge--Tutte type theorems and a matroidal result to the m-tree packing problem.
Bibtex entry:
AUTHOR | = | {Szab{\'o}, J{\'a}cint}, |
TITLE | = | {Packing trees with constraints on the leaf degree}, |
NOTE | = | {{\tt egres.elte.hu}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2007}, |
NUMBER | = | {TR-2007-04} |