The Maximum Number of Cycles in a Graph with Fixed Number of Edges

Abstract

The main topic considered is maximizing the number of cycles in a graph with given number of edges. In 2009, Király conjectured that there is constant c such that any graph with m edges has at most (1.4) m cycles. In this paper, it is shown that for sufficiently large m, a graph with m edges has at most (1.443) m cycles. For sufficiently large m, examples of a graph with m edges and (1.37) m cycles are presented. For a graph with given number of vertices and edges an upper bound on the maximal number of cycles is given. Also, exponentially tight bounds are proved for the maximum number of cycles in a multigraph with given number of edges, as well as in a multigraph with given number of vertices and edges.

Maximising the number of cycles in graphs with forbidden subgraphs

“…We define β(H) := 10n 0 . In a recent paper, Arman and Tsaturian [5] consider the maximum number of cycles in a graph with a fixed number of edges: They show that if G is an n-vertex graph with m edges, then…”

“…Another interesting problem was raised by Király [18] who asked for the maximum number of cycles in a graph with m edges can contain (without constraining the number of vertices); he conjectured an upper bound of 1.4 m cycles. In a recent paper Arman and Tsaturian [5] give an upper bound of 8.25 × 3 m/3 and a lower bound of 1.37 m , and conjecture that their upper bound is correct to within a (1 + o(1)) m factor. It would be interesting to consider the effect of adding the additional constraint of forbidding a subgraph.…”

Local connections and the larval competency strongly influence marine metapopulation persistence

“…In all species, we found many three‐node cycles (Table 2), with the only exception of S. punctatus , which was an acyclic network. Though the number of cycles appears large, these values are far from the maximum possible number of cycles in theoretical networks (Gerbner et al 2018, Arman and Tsaturian 2019) that might contribute to increased metapopulation persistence and stability (Artzy‐Randrup and Stone 2010).…”

Bounding the number of cycles in a graph in terms of its degree sequence

“…Figure 1: The lower bound construction of Arman and Tsaturian [2] giving simple graphs with Ω(κ m 1 ) cycles for κ 1 = (2 + 2 √ 2) 1/5 ≈ 1.3701.…”

“…Currently the best known lower bound is Ω(κ m 1 ) for κ 1 = (2 + 2 √ 2) 1/5 ≈ 1.3701 by Arman and Tsaturian [2] (see Figure 1).…”