We prove that there exists such that if G is a 5-critical graph, then where is the maximum number of vertex-disjoint cliques of size three or four.

Furthermore, the bound in Theorem 1.1 is tight since it is attained by infinitely many 5-critical graphs. In fact, in a subsequent paper [3], Kostochka.

Missing: Density | Show results with:Density

Kostochka and Yancey proved that every 5-critical graph G satisfies: |E(G)|≥9/4|V(G)|-5/4. A construction of Ore gives an infinite family of graphs meeting ...

We prove that every triangle‐free 4‐critical graph G $G$ satisfies e( G ) ≥ 5 v( G ) + 2 3 $e(G)\ge \frac{5v(G)+2}{3}$ . This result gives a unified proof ...

As a corollary, a triangle-free 5-critical graph G satisfies: |E(G)|>=(9/4 + e)|V(G)| - 5/4.

Bibliographic details on On the Minimum Edge-Density of 5-Critical Triangle-Free Graphs.

We realize that this solution is not perfect, but this is the best compromise we have been able to achieve without making the notation unnecessarily clumsy.

A graph is k-critical if it has chromatic number k, and removing any edge allows it to be properly (k − 1)-colored. Thus by requiring this of our triangle-free ...

On the Minimum Edge-Density of 5-Critical Triangle-Free Graphs, Electronic Notes in Discrete Mathematics 49 (2015), 667--673. Minor-Minimal Non-Projective ...