I'm wondering if it is possible to put a minimum bound on the chromatic number $X(G)$ of a graph with $n$ vertices. Namely, how would one show that $X(G) \geq n/(n-d)$, where $d$ is the minimum degree of a vertex in $G$?

I've been stuck on this for a while and would love some suggestions for a proof.



Let $\chi(G)$ denote the chromatic number of $G$, and $\delta(G)$ the minimum degree of $G$.

Say that we have colored $G$ with an optimal coloring (that is, a coloring in $\chi(G)$ colors); write $$ V(G):=V_1\overset{\cdot}{\cup}\cdots\overset{\cdot}{\cup}V_{\chi(G)}, $$ where $V_i$ is the set of vertices of $V$ which are colored with color $i$.

Note, then, that $$ \lvert V_1\rvert+\cdots+\lvert V_{\chi(G)}\rvert=n. $$ On the other hand, $$ \lvert V_1\rvert+\cdots+\lvert V_{\chi(G)}\rvert\leq\chi(G)\cdot m,\qquad m:=\max\{\lvert V_1\rvert,\ldots,\lvert V_{\chi(G)}\rvert\}. $$ Clearly, $m>0$; thus $n\leq\chi(G)\cdot m$ implies $$ \chi(G)\geq\frac{n}{m}. $$ Can you prove that $m\leq n-\delta(G)$?


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