Combinatorics (Fall 2010)/Random graphs

Erdős–Rényi Random Graphs

The probabilistic method

Coloring large-girth graphs

Definition

Let [math]\displaystyle{ G(V,E) }[/math] be an undirected graph. A cycle of length [math]\displaystyle{ k }[/math] in [math]\displaystyle{ G }[/math] is a sequence of distinct vertices [math]\displaystyle{ v_1,v_2,\ldots,v_{k} }[/math] such that [math]\displaystyle{ v_iv_{i+1}\in E }[/math] for all [math]\displaystyle{ i=1,2,\ldots,k-1 }[/math] and [math]\displaystyle{ v_kv_1\in E }[/math]. The girth of [math]\displaystyle{ G }[/math], dented [math]\displaystyle{ g(G) }[/math], is the length of the shortest cycle in [math]\displaystyle{ G }[/math]. The chromatic number of [math]\displaystyle{ G }[/math], denoted [math]\displaystyle{ \chi(G) }[/math], is the minimal number of colors which we need to color the vertices of [math]\displaystyle{ G }[/math] so that no two adjacent vertices have the same color. Formally, [math]\displaystyle{ \chi(G)=\min\{C\in\mathbb{N}\mid \exists f:V\rightarrow[C]\mbox{ such that }\forall uv\in E, f(u)\neq f(v)\} }[/math]. The independence number of [math]\displaystyle{ G }[/math], denoted [math]\displaystyle{ \alpha(G) }[/math], is the size of the largest independent set in [math]\displaystyle{ G }[/math]. Formally, [math]\displaystyle{ \alpha(G)=\max\{|S|\mid S\subseteq V\mbox{ and }\forall u,v\in S, uv\not\in E\} }[/math].

Theorem (Erdős 1959)

For all [math]\displaystyle{ k,\ell }[/math] there exists a graph [math]\displaystyle{ G }[/math] with [math]\displaystyle{ g(G)\gt \ell }[/math] and [math]\displaystyle{ \chi(G)\gt k\, }[/math].