55-构形与56-构形是可约的

Let G be a 4-chromatic maximal planar graph (MPG) with minimum degree of at least 4 and let C be an even-length cycle of G. If |f(C)|=2 for every f in some Kempe equivalence class of G, then we call C an unchanged bichromatic cycle (UBC) of G, and correspondingly G an unchanged bichromatic cycle maximal planar graph (UBCMPG) with respect to C, where f(C)={f(v)|v∈V(C)}. For an UBCMPG G with respect to an UBC C, the subgraph of G induced by the set of edges belonging to C and its interior (or exterior), denoted by G^C, is called a base-module of G; in particular, when the length of C is equal to four, we use C_4 instead of C and call G^(C_4 ) a 4-base-module. In this paper, we first study the properties of UBCMPGs and show that every 4-base-module G^(C_4 ) contains a 4-coloring under wihich C_4 is bichromatic and there are at least two bichromatic paths with different colors between one pair of diagonal vertices of C_4 (these paths are called module-paths). We further prove that every 4-base-module G^(C_4 ) contains a 4-coloring (called decycle coloring) for which the ends of a module-path are colored by distinct colors. Finally, based on the technique of the contracting and extending operations of MPGs, we prove that 55-configurations and 56-configurations are reducible by converting the reducibility problem of these two classes of configurations into the decycle coloring problem of 4-base-modules.