Abstract
The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d G (x) + d G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d G (u) ≥ k.
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Supported by the Ministry of Education and Science, Spain, and the European Regional Development Fund (ERDF) under Projects MTM2008-06620-C03-02 and MTM2008-05866-C03-01, and by the Catalonian Government under Project 1298 SGR2009 and Andalusian Government under Project P06-FQM-01649
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Balbuena, C., Márquez, A. & Portillo, J.R. A sufficient degree condition for a graph to contain all trees of size k . Acta. Math. Sin.-English Ser. 27, 135–140 (2011). https://doi.org/10.1007/s10114-011-9617-6
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DOI: https://doi.org/10.1007/s10114-011-9617-6