Abstract
A spanning subgraph H of a graph G is a 2-detour subgraph of G if for each x, y ∈ V(G), d H (x, y) ≤ d G (x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2-detour subgraph of the n-dimensional hypercube Q n has at least clog2 n · 2n edges.
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József Balogh: Research supported in part by NSF grants DMS-0302804, DMS-0603769 and DMS-0600303, UIUC Campus Reseach Board #06139 and #07048, and OTKA 049398.
Alexandr Kostochka: Research supported in part by NSF grants DMS-0400498 and DMS-0650784, and grant 06-01-00694 of the Russian Foundation for Basic Research.
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Balogh, J., Kostochka, A. On 2-Detour Subgraphs of the Hypercube. Graphs and Combinatorics 24, 265–272 (2008). https://doi.org/10.1007/s00373-008-0790-z
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DOI: https://doi.org/10.1007/s00373-008-0790-z