55- and 56-configurations are reducible

Abstract

Let $G$ be a 4-chromatic maximal planar graph (MPG) with the 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 $f(C)=\{f(v)| v\in V(C)\}$. 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 which $C_4$ 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.