20.500.12556/RUP-23441 Irregular graph labelings in Abelian groups Neregularna označevanja grafov v abelovih grupah Let G⃗ = (V,E) be a directed graph of order n. If there exists a mapping ψ from E(G⃗) to an Abelian group Γ such that if we define a mapping φ_ψ from V(G⃗) to Γ by φ_ψ(x) = ∑y ∈ N⁺(x)ψ(xy) − ∑y ∈ N⁻(x)ψ(yx),   (x ∈ V(G⃗)), then φψ is injective, then such a labeling ψ is called Γ-irregular. Recently it was showed that if n is large enough then G⃗ has a Γ-irregular labeling for any Γ such that |Γ| > (1 + ε)n (in the paper from Cichacz and Tuza from 2022). In this paper, we prove that if all weakly connected components of G⃗ are of size at least 4, then G⃗ has a Γ-irregular labeling for any finite group Γ such that |Γ| >= n + 5. Naj bo G⃗ = (V,E) usmerjen graf reda n. Če obstaja preslikava ψ iz E(G⃗) v abelovo grupo Γ, tako da je preslikava φ_ψ iz V(G⃗) v Γ definirana z φ_ψ(x) = ∑y ∈ N⁺(x)ψ(xy) − ∑y ∈ N⁻(x)ψ(yx),   (x ∈ V(G⃗)), injektivna, preslikavo ψ imenujemo Γ-neregularna. Nedavno je bilo pokazano, da če je n dovolj velik, potem ima G⃗ Γ-neregularno označanje za vsako grupo Γ, za katero velja |Γ| > (1 + ε)n (v članku Cichacza in Tuze iz leta 2022). V tem članku pokažemo, da če imajo vse šibko povezane komponente grafa G⃗ velikost vsaj 4, potem ima G⃗ Γ-neregularno označanje za vsako končno grupo Γ, za katero velja |Γ| ≥ n + 5. finite Abelian group directed graph zero-sum sets končne abelove grupe usmerjen graf množiče z ničelno vsoto true false true Založba Univerze na Primorskem Angleški jezik Slovenski jezik Članek v reviji 2026-08-11 11:23:07 2026-08-11 11:23:07 2026-08-12 03:07:39 0000-00-00 00:00:00 2026 0 0 10 str. no. 2, [article no.] P2.09 Vol. 26 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-03-31 51 1855-3974 10.26493/1855-3974.3380.cb8 AMC_Cichacz_2026.pdf AMC_Cichacz_2026.pdf 1 AC47343FA13DEBC61C1F7A32ED96CA17 a93aa99a7d90b5ca986376c96b206979ba972352d38bed63f36d2cf10f12e33b 36403d31-9567-11f1-9e8d-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=34354 Založba Univerze na Primorskem 0 0 0