We determine the maximum number of induced copies of a 5-cycle in a graph on n vertices for every n. Every extremal construction is a balanced iterated blow-up of the 5-cycle with the possible exception of the smallest level where for n = 8, the Möbius ladder achieves the same number of induced 5-cycles as the blow-up of a 5-cycle on 8 vertices.This result completes work of Balogh, Hu, Lidický, and Pfender [Eur. J. Comb. 52 ( 2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its use to small graphs.
We determine the maximum number of induced copies of a 5‐cycle in a graph on n $n$ vertices for every n $n$. Every extremal construction is a balanced iterated blow‐up of the 5‐cycle with the possible exception of the smallest level where for n
=
8 $n=8$, the Möbius ladder achieves the same number of induced 5‐cycles as the blow‐up of a 5‐cycle on eight vertices. This result completes the work of Balogh, Hu, Lidický, and Pfender, who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method, but we use a new and more sophisticated approach which allows us to extend its use to small graphs.
Let the diameter cover number, D t r (G), denote the least integer d such that under any r-coloring of the edges of the graph G, there exists a collection of t monochromatic subgraphs of diameter at most d such that every vertex of G is contained in at least one of the subgraphs. We explore the diameter cover number with two colors and two subgraphs when G is a complete multipartite graph with at least three parts. We determine exactly the value of D 2 2 (G) for all complete tripartite graphs G, and almost all complete multipartite graphs with more than three parts.
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