The Planar Tree Packing Theorem

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garc\'ia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.Comment: Full version of our SoCG 2016 pape

ρ\rho-Meson wave functions from nonlocal light-cone operators with definite twist

We introduce chiral-even and chiral-odd meson wave functions as vacuum-to-meson matrix elements of bilocal quark operators with well-defined (geometric) twist. Thereby, we achieve a Lorentz invariant classification of these distributions which differ from the conventional ones by explicitly taking into account the trace terms. The relations between conventional and new wave functions are given.Comment: 8 pages, REVTEX; corrected sign error

A Note on Wandzura-Wilczek Relations

Wandzura-Wilczek (WW) relations between matrix-elements of bilocal light-ray operators have recently regained interest in connection with off-forward scattering processes. Originally derived for matrix elements over leading-twist operators, their generalization to off-forward and exclusive processes gets complicated by the presence of higher-twist operators that are total derivatives of leading-twist ones and do not contribute to forward-scattering. We demonstrate that, for exclusive matrix-elements, the inclusion of these operators into WW-relations is essential for fulfilling constraints imposed by the conformal symmetry of massless QCD.Comment: 4 pages RevTe

Quantum Fiel Theoretic Treatment of the Non-Forward Compton Amplitude in the Generalized Bjorken Region

A quantum field theoretic treatment of the leading light-cone part of the virtual Compton amplitude is presented. The twist-decomposition of the operators is performed by a group-theoretic procedure respecting the Lorentz group O(3,1). The twist-2 contributions to the Compton amplitude are calculated and it is shown that the electromagnetic current is conserved for these terms. Relations between the amplitude functions associated to the symmetric and asymmetric part of the Compton amplitude are derived. These relations generalize the Callan-Gross and Wandzura-Wilczek relations of forward scattering for the non-forward Compton amplitude.Comment: 7 pages LATEX, 1 style file, DESY 00-045, Contribution to the Proceedings of `Loops and Legs in Quantum Field Theory', Bastei, Germany, April 2000, Nucl. Phys. B (Proc. Suppl.) (2000) to appea