An extremal result about vertex covers, attributed by Hajnal to Erdős and Gallai, is applied to prove the following: If n, k, and t are integers satisfying n ≥ k ≥ t ≥ 3 and k ≤ 2t - 2, and G is a graph with the minimum number of edges among graphs on n vertices with the property that every induced subgraph on k vertices contains a complete subgraph on t vertices, then every component of G is complete.
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Noble, Matt; Johnson, Peter; Hoffman, Dean; and McDonald, Jessica
"Application of an Extremal Result of Erdős and Gallai to the (n,k,t) Problem,"
Theory and Applications of Graphs: Vol. 4
, Article 1.
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol4/iss2/1