On the Total π Chromatic Number of Petal, Wing, and Tentacle Graph Families

The total pi chromatic number TπCN(G) of a graph G requires at least three colors to create a total coloring which provides every vertex with its distinct color combination between its own color and the colors of all neighboring vertices and attached edges. We determine TπCN(G) for eight graph families of geometric and biological character: the Jempiring, Butterfly, Lotus, Sunflower, Jellyfish, Snail, Octopus, and Lily graphs. The lower bound for each family comes from degree arguments which apply at the hub while the upper bound applies through a modular arithmetic coloring system that creates distinct color sets through explicit verification. Seven families satisfy TπCN(G)=Δ(G)+1 which demonstrates Type 1 behavior while the Jellyfish graph satisfies TπCN(J(n,n))=n+4 for all n≥2, a consequence of its diamond body which creates inter-hub constraints that force one extra color beyond the Type-1 bound. These findings extend the existing collection of graph families which have established total pi chromatic numbers.