For every simple graph $G$ and a proper coloring $C = \{ c_i | 1 \le i \le \chi \}$ with $\chi=\chi(G)$ colors (the minimum number of colors for a proper coloring). There is a vertex $v$ with color $c_p$ for some $p$ in $\{1, 2, \ldots, \chi\}$ such as for all colors $c_i$ with $i \ne p$ there is a neighbor of $v$ colored with $c_i$.
Where:
$\Delta(G)$ is the maximum degree of a vertex in $G$.
$\chi(G)$ is the minimum number of colors for a proper vertex coloring, also known as chromatic number
$d(v)$ is the degree of a vertex.
There is no limits to degree of $G$.
I know that:
$\chi(G) \le \Delta(G)+1$ (1)
If $d(v) = \Delta(G)$ (2) then the initial statement holds. (Proven false by $S_n$).
(1) only holds for complete and odd cycles, for all other graphs we have $\Delta(G)$ as upper bound.
Seems obvious since if there is no $v$ such $d(v) = \Delta(G)$, I could use less colors to color the graph. But don't know how to work with.