Discrete Mathematics & Theoretical Computer Science, Vol 14, No 1 (2012)

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α-Labelings and the Structure of Trees with Nonzero α-Deficit

Gunnar Brinkmann, Simon Crevals, Hadrien M\élot, Leanne Rylands, Eckhard Steffen


We present theoretical and computational results on α-labelings of trees. The theorems proved in this paper were inspired by the results of a computer investigation of α-labelings of all trees with up to 26 vertices, all trees with maximum degree 3 and up to 36 vertices, all trees with maximum degree 4 and up to 32 vertices and all trees with maximum degree 5 and up to 31 vertices. We generalise a criterion for trees to have nonzero α-deficit, and prove an unexpected result on the α-deficit of trees with a vertex of large degree compared to the order of the tree.

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