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Bounds for f(n) when n ≥ 13

Ascertain lower and upper bounds for f(n) for integers n ≥ 13, where f(n) is defined as the maximum over all nonempty n-vertex graphs G of str(G) + str(Ĝ), and str(G) denotes the minimum, over bijections f: V(G) → {1, ..., n}, of the maximum edge label max{f(u)+f(v) : uv ∈ E(G)}.

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Background

The function f(n) quantifies the extremal sum of strengths of a graph and its complement over all nonempty graphs of order n. The paper proves two lower bounds: f(n) ≥ 3n + floor(n/2) − 3 (Theorem 3.1) and a potentially improved bound f(n) ≥ 4n − 2√3 + √(8n − 7)/2 + 2 (Theorem 3.2).

Exact values are determined for n in the range 3 ≤ n ≤ 12 (Table 2). The authors explicitly state uncertainty for n ≥ 13, motivating the need to establish bounds in this regime.

References

As mentioned above, f (n) attains the bound presented in Theorem 3.1 for n = 3. Indeed, f (n) attains the same bound for n ∈ [4,12]. However, we do not know whether the case for n ≥ 13. Thus, we propose the next two problems.

Problem 3. Find good lower and upper bounds for f (n) when n ≥ 13.

— Ramsey theory and strength of graphs (2408.01475 - Ichishima et al., 2 Aug 2024) in Section 3, after Table 2