Rainbow Antimagic Coloring of Closed Sunflower Graphs and Their Multilayer Extensions

Abstract

The challenge in rainbow antimagic coloring is not to eliminate repeated edge weights, but to arrange them without obstructing rainbow paths. We address this problem for closed sunflower graphs and their multilayer extensions, with edge colors determined by sums of distinct vertex labels. Explicit bijective labelings yield upper bounds, while vertex degrees, distances, and regular subgraphs provide complementary lower bounds. The resulting estimates hold throughout the defining parameter ranges of both families, and the smallest closed sunflower graph has rainbow antimagic connection number seven. For cycle lengths of at least ten that are congruent to two modulo four, a separate construction reduces the gap between the bounds to two colors. In the multilayer family, an alternating labeling scheme gives an upper bound that grows linearly with the number of layers for each fixed cycle length and recovers the general closed sunflower construction in the two-layer case. All constructions include a complete count of distinct edge weights and a verification of rainbow connectivity. The general labelings admit rainbow spanning trees, whereas the improved construction uses explicit paths and fewer colors than a rainbow spanning tree would require. This distinction demonstrates how coordinated color reuse can strengthen constructive bounds beyond the spanning-tree approach.