Let mij (G) be the number of edges of graph G, connecting vertices of degrees i and j. Necessary and sufficient conditions are established on a symmetric matrix M of type ∆ ×∆ such that there is a tree T for which Mij = mij (T ) holds for all i, j.

The experimental results on the tt production cross section at the Tevatron are well described by the QCD contributions within the standard model, while the recently measured forward-backward asymmetry is larger than predicted within this framework. We consider light colored scalars appearing in a particular SU(5) grand unified theory model within the 45-dimensional Higgs representation. A virtue of the model is that it connects the presence of a light colored SU(2) singlet ({Delta}{sub 6}) and a color octet weak doublet ({Delta}{sub 1}) with bounds on the proton lifetime, which constrain the parameter space of both scalars. We find that both the total tt production cross section and the forward-backward asymmetry can be accommodated simultaneously within this model. The experimental results prefer a region for the mass of {Delta}{sub 6} around 400 GeV, while {Delta}{sub 1} is then constrained to have a mass around the TeV scale as well. We analyze possible experimental signatures and find that {Delta}{sub 6} associated top production could be probed in the tt+jets final states at Tevatron and the LHC.