组合数学 (Fall 2019)/Problem Set 3

Problem 3

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], it holds that [math]\displaystyle{ v\in D }[/math] or [math]\displaystyle{ v }[/math] is adjacent to a vertex 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?

Problem 4

"An edge [math]\displaystyle{ k }[/math]-coloring for [math]\displaystyle{ G(V,E) }[/math]" is a mapping [math]\displaystyle{ f:E\to[k] }[/math], where [math]\displaystyle{ E }[/math] is the edge set[math]\displaystyle{ E }[/math].

Problem 5

Let [math]\displaystyle{ G(V,E) }[/math] be a cycle of length [math]\displaystyle{ k\cdot n }[/math] and let [math]\displaystyle{ V=V_1\cup V_2\cup\dots V_n }[/math] be a partition of its [math]\displaystyle{ k\cdot n }[/math] vertices into [math]\displaystyle{ n }[/math] pairwise disjoint subsets, each of cardinality [math]\displaystyle{ k }[/math]. For [math]\displaystyle{ k\ge 11 }[/math] show that there must be an independent set of [math]\displaystyle{ G }[/math] containing precisely one vertex from each [math]\displaystyle{ V_i }[/math].

Problem 7.1

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: "An edge [math]\displaystyle{ k }[/math]-coloring for [math]\displaystyle{ G(V,E) }[/math]" is a mapping [math]\displaystyle{ f:E\to[k] }[/math], where [math]\displaystyle{ E }[/math] is the edge set [math]\displaystyle{ E }[/math].