Candidate Resolution of Open Problem 3.3: Defect Conservation and Exact Modular Edge-Irregularity Strength of Friendship Graphs

Abstract

We present a complete candidate proof for the unresolved friendship-graph family in Open Problem 3.3 of Koam, Ahmad, Bača, and Semaničová-Feňovčíková (AIMS Mathematics 8 (2023), 1475–1487). For the friendship graph F_n = K_1 ∨ nK_2, we claim that for every n ≥ 12 with n ≡ 0 (mod 4), es(F_n) = mes(F_n) = 2n + ceil(n/7). The lower bound is obtained from a new defect-conservation identity. Writing k = 2n+s and decomposing rim-edge weights below, inside, and above the spoke-weight band yields nonnegative slacks α, β, γ satisfying α+β+5γ = 7s−n, and hence s ≥ ceil(n/7). This gives the universal lower bound es(F_n) ≥ 2n+ceil(n/7) for every friendship graph. The upper bound is constructive: a finite low-weight kernel, an explicit reflected high-weight matching, and zero-rooted Schur partitions generated from Langford and near-Skolem sequences together produce 3n consecutive ordinary edge weights, which therefore represent all residues modulo 3n. All finite exceptional Schur certificates are included. An accompanying verifier checks the defect algebra, all exceptional certificates, complete representatives of the seven structural classes, and the construction identities through n = 1,000,000. The result is presented as a candidate proof pending independent expert verification and peer review.

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