Title: Chromatic Uniqueness and Equivalence of Generalized Fan Graphs
Authors: Hanan Hayder Mohammed
Volume: 10
Issue: 5
Pages: 179-182
Publication Date: 2026/05/28
Abstract:
This paper studies the chromatic uniqueness of the generalized fan graph F_(n,m)=O_n+ P_m, where n?1 and m?2. The objective is to determine the conditions under which F_(n,m) is chromatically unique and to classify its equivalence classes. By analyzing the chromatic polynomial, we show that uniqueness is restricted to the small cases: F_(n,m), n?2, m?3, with F_(1,3)?F_(2,2). For all other parameters, F_(n,m) is not unique, and infinite families of chromatic equivalences are derived, including F_(n,2) ?F_(1,m) and F_(n,3)?F_(2,m), m?n+1. This provides a complete characterization of the chromatic properties of generalized fan graphs and contributes to the broader theory of join-graph invariants.