Abstract
The concept of the k-pairable graphs was introduced by Zhibo Chen (On k-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter p(G), called the pair length of a graph G, as the maximum k such that G is k-pairable and p(G) = 0 if G is not k-pairable for any positive integer k. In this paper, we answer the two open questions raised by Chen in the case that the graphs involved are restricted to be trees. That is, we characterize the trees G with p(G) = 1 and prove that p(G □ H) = p(G) + p(H) when both G and H are trees.
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References
Z. Chen: On k-pairable graphs. Discrete Math. 287 (2004), 11–15.
N. Graham, R. C. Entringer, L. A. Székely: New tricks for old trees: maps and the pigeonhole principle. Amer. Math. Monthly 101 (1994), 664–667.
W. Imrich, S. Klavžar: Product Graphs: Structure and Recognition. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Chichester, 2000.
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Che, Z. On k-pairable graphs from trees. Czech Math J 57, 377–386 (2007). https://doi.org/10.1007/s10587-007-0066-4
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DOI: https://doi.org/10.1007/s10587-007-0066-4