Packing Chromatic Number of Graph Families Constructed from the Fan Graph

Packing coloring is particularly sensitive to graph distances, since even a simple structural modification may change whether a color can be reused. This paper investigates the packing chromatic number of several constructions derived from the fan graph, namely the subdivision fan S(Fn), the umbrella graph Ur,s, two barbell-type fan constructions, and the generalized q-fan chain CFm(q) formed by sequentially joining the centers of q copies of Fm. The known packing chromatic number of the ordinary fan is used only as a reference result. By combining explicit packing colorings with lower bound arguments based on independence numbers, graph diameters, color class capacities, and cross-component distances, we prove that (S(Fn)) = 4 for n 2 and determine the exact packing chromatic numbers of the umbrella and both barbell-type constructions. For CFm(q), we establish general lower and upper bounds for all m, q 2 by showing that a color 2 can occur on at most q/( 1) vertices. Moreover, when m 2q 2, the bounds coincide and yield (CFm(q)) = q(m/2 + 2) t=1q1q/t. These results show that the packing chromatic behavior of fan-derived graphs is governed by the distance structure created by the underlying graph operation, particularly the extent to which low colors can be reused across different fan copies.