The Structure of the 2-factor Transfer Digraph common for Thin Cylinder, Torus and Klein Bottle Grid Graphs

2022/12/28 by Jelena Đokić, Đokić, Jelena, Ksenija Doroslovački +3
Computer Science · #05A15 #05C30 #05C38 #05C50 #05C85 #Advanced Graph Theory Research #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #Interconnection Networks and Systems

paper · pdf · doi:10.48550/arxiv.2212.13779

openalex publication_date 2022/12/28 · openalex created_date 2023/01/06 · openalex updated_date 2026/08/04

Abstract

We prove that the transfer digraph \cal D^*_C,m needed for the enumeration of 2-factors in the thin cylinder TnC_m(n), torus TG_m(n) and Klein bottle KB_m(n) (all grid graphs of the fixed width m and with m ⋅ n vertices), when m is odd, has only two components of order 2^m-1 which are isomorphic. When m is even, \cal D^*_C,m has \lfloor (m)/(2) \rfloor + 1 components which orders can be expressed via binomial coefficients and all but one of the components are bipartite digraphs. The proof is based on the application of recently obtained results concerning the related transfer digraph for linear grid graphs (rectangular, thick cylinder and Moebius strip).

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