Exploring Adjacency Recognizable Colorings in Graphs: A New Variant of Adjacency Codes

Let G be a nontrivial connected graph and c : V(G) → {1, 2, ..., k} be a coloring of G, where adjacent vertices may be colored the same. For any vertex v of G, the adjacency code adc(v) of v with respect to c is defined as the ordered k-tuple , where is the number of vertices adjacent to v that are colored i for 1 ≤ i ≤ k. The coloring c is called adjacency recognizable if distinct vertices have distinct adjacency codes, and the adjacency recognition number of G, denoted an(G), is the minimum po