随机图同态与对数度
Random Graph-Homomorphisms and Logarithmic Degree
摘要：
两个图之间的图同态是从一个图的顶点集到另一个图的顶点集的映射，其将边映射到边。本文研究了从图G到无限直线Z的一致选择同态的值域。证明了如果G的最大度是“次同态”，则这种同态的值域是超常数的。此外，一些例子表明，也许对于超对数度的图，一个典型的同态的范围是有界的。特别是，一个尖锐的过渡示出了一个特定的家庭的图C_{n,k}（这是张量积的n-循环和完全图，自循环，大小为k）。也就是说，给定任何趋于无穷大的函数psi（n），C_{n,k}的典型同态的范围对于k = 2 log（n）- psi（n）是超常数，并且对于k = 2 log（n）+ psi（n）是3。
英文摘要：
A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is `sub-logarithmic', then the range of such a homomorphism is super-constant. Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs C_{n,k} (which is the tensor product of the n-cycle and a complete graph, with self-loops, of size k). That is, given any function psi(n) tending to infinity, the range of a typical homomorphism of C_{n,k} is super-constant for k = 2 log(n) - psi(n), and is 3 for k = 2 log(n) + psi(n).
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