alpha-Labelings and the Structure of Trees with Nonzero alpha-Deficit
Gunnar Brinkmann (1) (2), Simon Crevals (1) (2), Hadrien Melot (1) (2), Leanne Rylands (3), Eckhard Steffen (4)
(1) Department of Applied Mathematics and Computer Science [Ghent]
(2) Department of Mathematics, Computer Science and Statistics [Ghent]
(3) School of Computing, Engineering and Mathematics [Sydney]
(4) Paderborn Institute for Advanced Studies in Computer Science and Engineering
Abstract
We present theoretical and computational results on alpha-labelings of trees. The theorems proved in this paper were inspired by the results of a computer investigation of alpha-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 alpha-deficit, and prove an unexpected result on the alpha-deficit of trees with a vertex of large degree compared to the order of the tree.
