Let $G$ be a simple connected finite graph and let $v \ge 4$ be the number of vertices, $E$ the number of edges, $\chi(G)$ the chromatic number , $\omega(G)$ the clique number and $\Delta$ the maximum degree of G. Let $C(G)$ be a function of $G$ which is equal to $3$ when $G$ contains cycles of odd length. Let $ {(d_1,d_2,...)}$ be the degree sequence of G.
Let $i > 4$ be the smallest integer such that
$$ \left\lceil \frac{(0.5\sum\limits_{j=1}^i d_j) -i}{i-3} \right\rceil < \frac{i}{2}$$
and
$$ 2\frac{(0.5\sum\limits_{j=1}^i d_j) -i}{i-3} < \Delta $$
Take $i = v$ if a minimum value does not exist.
Let
$ B = 4 $ if $$ 1.5\leq\frac{(0.5\sum\limits_{j=1}^i d_j) -i}{i-3} < 2 $$
Otherwise
$$ B = \left\lfloor 2\frac{(0.5\sum\limits_{j=1}^i d_j) -i}{i-3} \right\rfloor $$
Then
$$\chi(G) \le \max\left\{C(G) \ , B,\ \omega(G)\right\}$$