An Inertial Lower Bound for the Chromatic Number of a Graph

Let χ(G) and χf (G) denote the chromatic and fractional chromatic numbers of a graph G, and let (n+, n0, n−) denote the inertia of G. We prove that: 1 + max ( n+ n− , n− n+ ) 6 χ(G) and conjecture that 1 + max ( n+ n− , n− n+ ) 6 χf (G). We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with a discussion of asymmetry between n+ and n−, including some Nordhaus-Gaddum bounds for inertia.