Geodesic
Spanning tree congestion of rook's graphs
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 753-761

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Let G be a connected graph and T be a spanning tree of G. For e ∈ E(T), the congestion of e is the number of edges in G joining the two components of T - e. The congestion of T is the maximum congestion over all edges in T. The spanning tree congestion of G is the minimum congestion over all its spanning trees. In this paper, we determine the spanning tree congestion of the rook's graph Kₘ ☐ Kₙ for any m and n.
Keywords: spanning tree congestion, Rook's graph 
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Kozawa, Kyohei; Otachi, Yota. Spanning tree congestion of rook's graphs. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 753-761. http://geodesic.mathdoc.fr/item/DMGT_2011_31_4_a8/