### A refined Tur'an theorem

#### Abstract

Let *G* = (*V*, *E*) be a finite undirected graph with vertex set *V*(*G*) of order |*V*(*G*)| = *n* and edge set *E*(*G*) of size |*E*(*G*)| = *m*. Let *Δ* = *d*_{1} ≥ *d*_{2} ≥ ⋯ ≥ *d*_{n} = *δ* be the degree sequence of the graph *G*. A *clique* in a graph *G* is a complete subgraph of *G*. The *clique number* of a graph *G*, denoted by *ω*(*G*), is the order of a maximum clique of *G*. In 1907 Mantel proved that a triangle-free graph with *n* vertices can contain at most ⌊*n*^{2}/4⌋ edges. In 1941 Tur'an generalized Mantel’s result to graphs not containing cliques of size *r* by proving that graphs of order *n* that contain no induced *K*_{r} have at most (1 − 1/*r* − 1)*n*^{2}/2 edges. In this paper, we give new bounds for the maximum number of edges in a *K*_{r}-free graph *G* of order *n*, minimum degree *δ*, and maximum degree *Δ*. We show that, for the families of graphs having the above properties, our bounds are slightly better than the more general bounds of Tur'an.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2024.12.1.2

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