# Label distance number and chromatic number of a graph

Let $$n\in\mathbb{N}$$ be a positive integer, and let $$[n] = \{1,\ldots,n\}$$. We set $$S_n$$ for the set of all bijections $$\varphi:[n]\to [n]$$.

Let $$G= ([n], E)$$ be a simple, undirected graph, and let $$\varphi\in S_n$$ We define the label distance number of $$G$$ in the following way: $$\lambda(G) = \min\big\{\max\{|\varphi(v)-\varphi(w)|:\{v,w\}\in E\}: \varphi\in S_n\big\}.$$

Conjecture: if $$n$$ is a positive integer, and $$G, H$$ are graphs with $$V(G) = V(H) = [n]$$ and $$\chi(G) < \chi(H)$$, then $$\lambda(G)\leq \lambda(H)$$.

Is this conjecture true?

I would also be interested in knowing whether there is an established name for what I call label distance number.