Inverting the Turán problem with chromatic number

• Published in: Discrete mathematics Vol. 344; no. 9; p. 112517
• Main Authors: Zhu, Xiutao, Chen, Yaojun
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2021
• Subjects: Chromatic number; Extremal graph; Turán problem; Turán problem; Chromatic number; Extremal graph
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2021.112517

For a graph G and a family of graphs H, the Turán number ex(G,H) is defined to be the maximum number of edges among all H-free subgraphs of G. Inverting this problem, Briggs and Cox (2019) [5] studied the extremal function εH(k)=sup{e(G)|ex(G,H)<k}, where e(G) is the size of G, and suggested to investigate the extremal function φH(k)=sup{χ(G)|ex(G,H)<k}, where χ(G) denotes the chromatic number of G. Let Kn be a complete graph of order n and H a given graph. In this paper, we establish a tight general upper bound for φH(k) and conjecture φH(k)=max{n|ex(Kn,H)<k} for H≠2K2. We also confirm this conjecture for many instances of H.