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

1997
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Electronic Journal of Combinatorics
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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

doi:10.37236/1342
fatcat:enjcwxszyjcwzgrhv4sx4oxizy