2025/01/14 by Binzhou Xia, Xia, Binzhou, Junyang Zhang +3 · 1 citation
Computer Science · Engineering · #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2501.08101
openalex publication_date 2025/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A subset C of the vertex set V of a graph Γ is called a perfect code in Γ if every vertex in V∖ C is adjacent to exactly one vertex in C. Given a group G and a subgroup H of G, a subgroup A of G containing H is called a perfect code of the pair (G,H) if there exists a coset graph Cos(G,H,U) such that the set of left cosets of H in A is a perfect code in Cos(G,H,U). In particular, A is called a perfect code of G if A is a perfect code of the pair (G,1). In this paper, we give a characterization of A to be a perfect code of the pair (G,H) under the assumption that H is a perfect code of G. As a corollary, we derive an additional sufficient and necessary condition for A to be a perfect code of G. Moreover, we establish conditions under which A is not a perfect code of (G,H), which is applied to construct infinitely many counterexamples to a question posed by Wang and Zhang [J.~Combin.~Theory~Ser.~A, 196 (2023) 105737]. Furthermore, we initiate the study of determining which maximal subgroups of Sn are perfect codes.