Two Upper Bounds on the Chromatic Number

Processor cache memory management is a challenging issue as it deeply impact performances and power consumption of electronic devices. It has been shown that allocating data structures to memory for a given application (as MPEG encoding, filtering or any other signal processing application) can be modeled as a minimum k-weighted graph coloring problem, on the so-called conflict graph. The graph coloring problem plays an important role as a particular case of the minimum weighted graph coloring problem, and providing upper bounds on the minimum number of colors to be used is an important issue for addressing these memory allocation problems. A coloring of graph G = (X,U) is a function F : X → N∗; where each node in X is allocated an integer value that is called a color. A proper coloring satisfies F (u) 6= F (v) for all (u, v) ∈ U [2]. The chromatic number of G, denoted by χ(G), is the smallest number of colors involved in any proper coloring. Determining χ(G) for any graphG is aNP-hard problem [1] however there are some well known particular cases: χ(G) = 1 if and only if G is a totally disconnected graph, χ(G) = 2 for any exactly bipartite graphs (including trees and forests) and χ(G) = |X| if G is complete.