On extensions of a conjecture of Gallai (Q1362102)

The authors show that in a \(k\)-critical graph \((k\geq 3)\) with more than \(k\) vertices, there exists an edge that is in fewer than \(2k/5\) cliques of size \(k-1\). They use this result to prove that if \(G\) is a \(k\)-critical graph of order \(n>k\geq 3\), then \(G\) contains fewer than \(n-3k/5+2\) complete subgraphs of order \(k-1\).