In a graph Γ, a perfect code C is an independent set of the vertex set V(Γ) such that every vertex outside C is adjacent to exactly one vertex in C. This notion generalizes classical perfect error-correcting codes and has been extensively studied in Cayley graphs. This paper extends the concept of perfect codes in Cayley graphs to Bi-Cayley graphs and presents a criterion for determining the existence of subgroup perfect codes in Bi-Cayley graphs. Let G be a finite group and let H, K≤G. The pair (H,K) is called a subgroup perfect code of (G,G) if there exists a bi-Cayley graph over G such that H0∪K1 forms a perfect code. The main result provides a necessary and sufficient condition for (H,K) to be such a code: there exists a subset S⊆G such that H has an inverse-closed transversal in G∖((S-1*K)⊔H) and K has an inverse-closed transversal in G∖((S*H)⊔K), where S*H denotes the disjoint union of left cosets. The result offers a systematic method to construct and verify perfect codes in bi-Cayley graphs, and reveals how group structure determines combinatorial domination in symmetric graphs.

Shi Weiyue. Research on the Existence Criterion for Subgroup Perfect Codes in Bi-Cayley Graphs. Academic Journal of Mathematical Sciences (2026), Vol. 7, Issue 1: 39-44. https://doi.org/10.25236/AJMS.2026.070106.

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