Constructions of universalized Sierpiński graphs based on labeling manipulations

Sierpiński graphs are known to be graphs with self-similar structures, and their various properties have been investigated until now. Besides, they are known to be isomorphic to WK-recursive networks which have been proposed as interconnection networks because of their nice extendability. As a generalization (resp., variant) of Sierpiński graphs, generalized Sierpiński graphs (resp., extended Sierpiński graphs) have been introduced. In this talk, we newly define a larger graph class which includes both generalized Sierpiński graphs and extended Sierpiński graphs, and moreover uneven graphs with self-similar structures such as Fobonacci trees. Let G be a simple undirected graph which may have a self-loop. The universalized Sierpiński graph Υ(G, n) is defined to be the graph with the vertex set consisting of all n-tuple (v1, v2, . . . , vn) where vi ∈ V (G) for 1 ≤ i ≤ n and vi 6= vi+1 if vi has no self-loop, and in which two vertices (u1, u2, . . . , un) and (v1, v2, . . . , vn) are adjacent if and only if there exists an integer j such that ui = vi for 1 ≤ i < j, ujvj ∈ E(G), and uj+` = vj, vj+` = uj for 1 ≤ ` ≤ n − j if both u and v have a self-loop, uj+` = vj, vj+` = uj (resp., uj+` = uj, vj+` = vj) for odd (resp., even) 1 ≤ ` ≤ n − j otherwise. In particular, when G is a compete graph with a self-loop at each vertex, Υ(G, n) is a Sierpiński graph. Besides, if every vertex of G has a self-loop (resp., G is a complete graph), then Υ(G, n) corresponds to a generalized Sierpiński graph (resp., extended Sierpiński graph). To construct Υ(G, n) based on the definition, we need to combine |V (G)| copies of subgraphs of Υ(G, n − 1). We then present an inductive algorithm whose essential parts consist of edge-labelings and vertex-labelings, to construct Υ(G, n) from the original graph G. An advantage of such an algorithmic construction of Υ(G, n) is to be able to investigate the structure of Υ(G, n) directly from that of Υ(G, n−1). We also present structural properties of universalized Sierpński graphs such as connectivity, vertex-colorings, edge-colorings, total colorings, hamiltonicity, and factorizations.