  Given a graphic degree sequence $D$, let $\chi(D)$ (respectively $\omega(D)$,
$h(D)$, and $H(D)$) denote the maximum value of the chromatic number
(respectively, the size of the largest clique, largest clique subdivision, and
largest clique minor) taken over all simple graphs whose degree sequence is
$D$. It is proved that $\chi(D)\le h(D)$. Moreover, it is shown that a
subdivision of a clique of order $\chi(D)$ exists where each edge is subdivided
at most once and the set of all subdivided edges forms a collection of disjoint
stars. This bound is an analogue of the Hajos Conjecture for degree sequences
and, in particular, settles a conjecture of Neil Robertson that degree
sequences satisfy the bound $\chi(D)\le H(D)$ (which is related to the Hadwiger
Conjecture). It is also proved that $\chi(D)\le {6/5}\omega(D)+{3/5}$ and that
$\chi(D) \le {4/5}\omega(D) + {1/5}\Delta(D) + 1$, where $\Delta(D)$ denotes
the maximum degree in $D$. The latter inequality is a strengthened version of a
conjecture of Bruce Reed. All derived inequalities are best possible.
