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Exact values of f(n) for n ≥ 13

Determine the exact values of f(n) for all integers n ≥ 13, where f(n) is defined as the maximum, over all nonempty n-vertex graphs G, of str(G) + str(Ĝ), with str(G) being 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

Building on established lower bounds and exact values for small n, the authors note that f(n) achieves their lower bound for n ≤ 12 but explicitly acknowledge uncertainty for larger n. They thus pose the problem of determining exact values for n ≥ 13.

This problem is central to extending the understanding of the extremal behavior of the strength parameter under complementarity beyond the currently resolved range.

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 4. Determine the exact values of f (n) for integers n ≥ 13.

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