The Median Stabilization Degree of a Median Algebra

The median stabilization degree (msd, for short) of a median algebra measures the largest possible number of steps needed to generate a subalgebra with an arbitrary set of generators. We determine the value of msd of a graphic n-cube Qn and we derive an estimation of msd for the natural median operator of Rn which is sharp up to one or two units. Interestingly, msd of Qn and of Rn grows like log1.5n. Finally, we characterize median algebras and median graphs of msd 1 in terms of forbidden subspaces.     

没有4至6-圈的平面图是(1,0,0)-可染的

设d₁, d₂,..., d_k 是k个非负整数. 若图G=(V,E) 的顶点集V可剖分成k个子集V₁, V₂,..., V_k,使得对i=1, 2,..., k 由V_i 所导出的子图G[V_i] 的最大度至多为d_i, 则称G是(d₁, d₂,..., d_k)-可染的. 著名的Steinberg 猜想断言, 每个既没有4-圈又没有5-圈的平面图是(0, 0, 0)-可染的. 对此猜想已经证明每个没有4 至7-圈的平面图是(0, 0, 0)-可染的, 但还没有发现有人证明每个没有4 至6-圈的平面图是(0, 0, 0)-可染的. 本文证明没有4 至6-圈的平面图是(1, 0, 0)-可染的.