Geodesic
On $k$-pairable graphs from trees
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 377-386.

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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 $p(G \square H)=p(G)+p(H)$ when both $G$ and $H$ are trees.
Classification : 05C05, 05C60, 05C75, 68R10
Keywords: $k$-pairable graph; pair length; Cartesian product; $G$-layer; tree 
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     title = {On $k$-pairable graphs from trees},
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     pages = {377--386},
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2007__57_1_a27/}
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Che, Zhongyuan. On $k$-pairable graphs from trees. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 377-386. http://geodesic.mathdoc.fr/item/CMJ_2007__57_1_a27/