More Tales of Hoffman: Bounds for the Vector Chromatic Number of a Graph

Let χ(G) denote the chromatic number of a graph and χv(G) denote the vector chromatic number. For all graphs χv(G) ≤ χ(G) and for some graphs χv(G) ≪ χ(G). Galtman proved that Hoffman's well-known lower bound for χ(G) is in fact a lower bound for χv(G). We prove that two more spectral lower bounds for χ(G) are also lower bounds for χv(G). We then use one of these bounds to derive a new characterization of χv(G).