On the Number of 4-Edge Paths in Graphs With Given Edge Density

Abstract

We investigate the number of 4-edge paths in graphs with a fixed number of vertices and edges.An asymptotically sharp upper bound is given to this quantity. The extremal construction is the quasi-star or the quasi-clique graph, depending on the edge density. An easy lower bound is alsoproved. This answer resembles the classic theorem of Ahlswede and Katona about the maximal number of 2-edge paths, and a recent theorem of Kenyon, Radin, Ren and Sadun about k-edge stars.

On the maximum number of copies of H in graphs with given size and order

“…Nagy [15] determined the asymptotic value of ex(n, e, P 4 ). Answering a question of his (Question 3 in [15]) in the affirmative, we prove the following theorem. Theorem 1.2.…”

On a Conjecture of Nagy on Extremal Densities

“…To restate the basic one: given a small graph G on v vertices, we would like to know which large graphs H asymptotically maximise N (G, H), the number of unlabelled copies of G in H, where H runs over all graphs on n vertices and edge density β. Interest in this problem was revitalised by its connection with graphons, and subsequently by the work of Nagy [11], who solved it for G = P 4 . Nagy's result is that for P 4 , as for P 2 , the maximiser is first a quasi-star and then a quasi-clique, with the flip occurring this time at β = 0.0865..., instead of 1 2 .…”

Threshold Graphs Maximize Homomorphism Densities

“…k We also provide an example in Theorem 4.6 of a graph H which for some c ∈ [0, 1] is optimized on neither the quasi-star nor the quasi-clique, disproving a conjecture of Nagy in [12]. To do so, we find a class of graphs and a graph G that does better for c sufficiently small.…”

“…This result was later generalized to k-stars, showing that for any c ∈ [0, 1] the number of homomorphisms from the k-star is maximized when G is the quasi-star or quasi-clique for small k [8], and shortly after for all k ≥ 2 [13]. The question was also studied in the case where H is the 4-edge path, and again it was shown that the optimizing graph is always either the quasi-star and quasi-clique for all densities [12]. Using a local move, we show that the maximum is always attained on a threshold graph, a class of graphs containing both the quasi-clique and quasi-star (See Section 2).…”

“…Instead of considering the lim sup over all graphs, we can work directly with graphons, which are graph limit objects, and find the graphon that maximizes the number of homomorphisms from H [10]. This approach is employed in the results of Nagy as well as Reiher and Wagner [12,13]. Threshold graphs may be more convenient to work with as their limits are one-dimensional, as opposed to graphons which are two dimensional [5].…”