### An improved bound on the minimal number of edges in color-critical graphs

Michael Krivelevich
1997 Electronic Journal of Combinatorics
A graph $G$ is $k$-color-critical (or simply $k$-critical) if $\chi(G)=k$ but $\chi(G') < k$ for every proper subgraph $G'$ of $G$, where $\chi(G)$ denotes the chromatic number of $G$. Consider the following problem: given $k$ and $n$, what is the minimal number of edges in a $k$-critical graph on $n$ vertices? It is easy to see that every vertex of a $k$-critical graph $G$ has degree at least $k-1$, implying $|E(G)|\geq {{k-1}\over {2}}|V(G)|$. Gallai improved this trivial bound to $|E(G)|\geq more » ... ound to$|E(G)|\geq {{k-1}\over {2}}+{{k-3}\over {2(k^2-3)}}|V(G)|$for every$k$-critical graph$G$(where$k\geq 4$), which is not a clique$K_k$on$k$vertices. In this note we strengthen Gallai's result by showing Theorem Suppose$k\geq 4$, and let$G=(V,E)$be a$k$-critical graph on more than$k$vertices. Then$ |E(G)|\geq ({{k-1}\over {2}}+{{k-3}\over {2(k^2-2k-1)}})|V(G)| \$