On Graph Associations

We introduce a notion of vertex association and consider sequences of these associations. This allows for slick proofs of a few known theorems as well as showing that for any induced subgraph H of G, χ(G) ≤ χ(H) + 12 (ω(G) + |G| − |H| − 1). As a special case of this, we have χ(G) ≤ ⌈ ω(G)+τ(G) 2 ⌉ (here χ(G) denotes the chromatic number, ω(G) the clique number and τ(G) the vertex cover number), which is a generalization of the Nordhaus-Gaddum upper bound. In addition, this settles a conjecture of Reed that χ(G) ≤ ⌈ ω(G)+∆(G)+1 2 ⌉ in the case when δ(G) ≤ ω(G). 1 Definitions and Basic Properties All graphs will be assumed finite and simple. We let |G| denote the order of G, s(G) the size of G, χ(G) the chromatic number, ω(G) the clique number, τ(G) the vertex cover number, ∆(G) the maximum degree, δ(G) the minimum degree, dG(x) the degree of x in G, and NG(x) the set of neighbors of x in G. Definition 1.1. Given a graph G and non-adjacent vertices a and b, we write G/[a, b] for the graph obtained from G by associating (i.e., identifying) a and b into a single vertex [a, b] and discarding multiple edges. Proposition 1.2. Let G be a graph and a, b, x ∈ V (G) with a 6∈ NG(b). Then dG/[a,b](x) =  dG(x)− 1 if x ∈ NG(a) ∩NG(b), dG(a) + dG(b)− |NG(a) ∩NG(b)| if x ∈ {a, b}, dG(x) otherwise. Proof. Immediate from the definitions. The content of the following proposition is that the operations of vertex removal and association commute.