概率论 (Summer 2013)/Problem Set 3

Problem 1

A set of vertices [math]\displaystyle{ D\subseteq V }[/math] of graph [math]\displaystyle{ G(V,E) }[/math] is a dominating set if for every [math]\displaystyle{ v\in V }[/math], either [math]\displaystyle{ v\in D }[/math] or one of its neighbour is in [math]\displaystyle{ D }[/math]. The problem of computing minimum dominating set is NP-hard.

• Prove that for every [math]\displaystyle{ d }[/math]-regular graph with [math]\displaystyle{ n }[/math] vertices, there exists a dominating set with size at most [math]\displaystyle{ \frac{n(1+\ln(d+1))}{d+1} }[/math].

• Try to obtain an upper bound for the size of dominating set using Lovász Local Lemma. Is it better or worse than previous one? Why?

Problem 2

Let [math]\displaystyle{ H(W,F) }[/math] be a graph and [math]\displaystyle{ n\gt |W| }[/math] be an integer. It is known that for some graph [math]\displaystyle{ G(V,E) }[/math] such that [math]\displaystyle{ |V|=n }[/math], [math]\displaystyle{ |E|=m }[/math], [math]\displaystyle{ G }[/math] does not contain [math]\displaystyle{ H }[/math] as a subgraph. Prove that for [math]\displaystyle{ k\gt \frac{n^2\ln n}{m} }[/math], there is an edge [math]\displaystyle{ k }[/math]-coloring for [math]\displaystyle{ K_n }[/math] that [math]\displaystyle{ K_n }[/math] contains no monochromatic [math]\displaystyle{ H }[/math].

Remark: Let [math]\displaystyle{ E=\binom{V}{2} }[/math] be the edge set of [math]\displaystyle{ K_n }[/math]. "An edge [math]\displaystyle{ k }[/math]-coloring for [math]\displaystyle{ K_n }[/math]" is a mapping [math]\displaystyle{ f:E\to[k] }[/math].

Problem 3

Let [math]\displaystyle{ G(V,E) }[/math] be a cycle of length [math]\displaystyle{ 4n }[/math] and let [math]\displaystyle{ V=V_1\cup V_2\cup\dots V_n }[/math] be a partition of its [math]\displaystyle{ 4n }[/math] vertices into [math]\displaystyle{ n }[/math] pairwise disjoint subsets, each of cardinality 4. Is it true that there must be an independent set of [math]\displaystyle{ G }[/math] containing precisely one vertex from each [math]\displaystyle{ V_i }[/math]? (Prove or supply a counter example.)

Hint: you can use Lovász Local Lemma.