Shortest Reconfiguration of Colorings Under Kempe Changes

International audienc

The Complexity of Change

Many combinatorial problems can be formulated as "Can I transform configuration 1 into configuration 2, if certain transformations only are allowed?". An example of such a question is: given two k-colourings of a graph, can I transform the first k-colouring into the second one, by recolouring one vertex at a time, and always maintaining a proper k-colouring? Another example is: given two solutions of a SAT-instance, can I transform the first solution into the second one, by changing the truth value one variable at a time, and always maintaining a solution of the SAT-instance? Other examples can be found in many classical puzzles, such as the 15-Puzzle and Rubik's Cube. In this survey we shall give an overview of some older and more recent work on this type of problem. The emphasis will be on the computational complexity of the problems: how hard is it to decide if a certain transformation is possible or not?Comment: 28 pages, 6 figure

Reconfiguration of Colorings in Triangulations of the Sphere

In 1973, Fisk proved that any 4-coloring of a 3-colorable triangulation of the 2-sphere can be obtained from any 3-coloring by a sequence of Kempe-changes. On the other hand, in the case where we are only allowed to recolor a single vertex in each step, which is a special case of a Kempe-change, there exists a 4-coloring that cannot be obtained from any 3-coloring. In this paper, we present a linear-time checkable characterization of a 4-coloring of a 3-colorable triangulation of the 2-sphere that can be obtained from a 3-coloring by a sequence of recoloring operations at single vertices. In addition, we develop a quadratic-time algorithm to find such a recoloring sequence if it exists; our proof implies that we can always obtain a quadratic length recoloring sequence. We also present a linear-time checkable criterion for a 3-colorable triangulation of the 2-sphere that all 4-colorings can be obtained from a 3-coloring by such a sequence. Moreover, we consider a high-dimensional setting. As a natural generalization of our first result, we obtain a polynomial-time checkable characterization of a k-coloring of a (k-1)-colorable triangulation of the (k-2)-sphere that can be obtained from a (k-1)-coloring by a sequence of recoloring operations at single vertices and the corresponding algorithmic result. Furthermore, we show that the problem of deciding whether, for given two (k+1)-colorings of a (k-1)-colorable triangulation of the (k-2)-sphere, one can be obtained from the other by such a sequence is PSPACE-complete for any fixed k ? 4. Our results above can be rephrased as new results on the computational problems named k-Recoloring and Connectedness of k-Coloring Reconfiguration Graph, which are fundamental problems in the field of combinatorial reconfiguration

Kempe equivalence of 44-critical planar graphs

Answering a question of Mohar from 2007, we show that for every $4$-critical planar graph, its set of $4$-colorings is a Kempe class.Comment: 7 pages, 2 figures; fixed some typo

Kempe Equivalent List Colorings

An $\alpha,\beta$-Kempe swap in a properly colored graph interchanges the colors on some component of the subgraph induced by colors $\alpha$ and $\beta$. Two $k$-colorings of a graph are $k$-Kempe equivalent if we can form one from the other by a sequence of Kempe swaps (never using more than $k$ colors). Las Vergnas and Meyniel showed that if a graph is $(k-1)$-degenerate, then each pair of its $k$-colorings are $k$-Kempe equivalent. Mohar conjectured the same conclusion for connected $k$-regular graphs. This was proved for $k=3$ by Feghali, Johnson, and Paulusma (with a single exception $K_2\square K_3$, also called the 3-prism) and for $k\ge 4$ by Bonamy, Bousquet, Feghali, and Johnson. In this paper we prove an analogous result for list-coloring. For a list-assignment $L$ and an $L$-coloring $\varphi$, a Kempe swap is called $L$-valid for $\varphi$ if performing the Kempe swap yields another $L$-coloring. Two $L$-colorings are called $L$-equivalent if we can form one from the other by a sequence of $L$-valid Kempe swaps. Let $G$ be a connected $k$-regular graph with $k\ge 3$. We prove that if $L$ is a $k$-assignment, then all $L$-colorings are $L$-equivalent (again with a single exception $K_2 \square K_3$). When $k\ge 4$, the proof is completely self-contained, so implies an alternate proof of the result of Bonamy et al. Our proofs rely on the following key lemma, which may be of independent interest. Let $H$ be a graph such that for every degree-assignment $L_H$ all $L_H$-colorings are $L_H$-equivalent. If $G$ is a connected graph that contains $H$ as an induced subgraph, then for every degree-assignment $L_G$ for $G$ all $L_G$-colorings are $L_G$-equivalent.Comment: 29 pages, 12 figures; second version extends the main result to cliques, which were previously excluded; third version incorporates reviewer feedback; to appear in Combinatoric

Reconfiguring Graph Colorings

Graph coloring has been studied for a long time and continues to receive interest within the research community \cite{kubale2004graph}. It has applications in scheduling \cite{daniel2004graph}, timetables, and compiler register allocation \cite{lewis2015guide}. The most popular variant of graph coloring, k-coloring, can be thought of as an assignment of $k$ colors to the vertices of a graph such that adjacent vertices are assigned different colors. Reconfiguration problems, typically defined on the solution space of search problems, broadly ask whether one solution can be transformed to another solution using step-by-step transformations, when constrained to one or more specific transformation steps \cite{van2013complexity}. One well-studied reconfiguration problem is the problem of deciding whether one k-coloring can be transformed to another k-coloring by changing the color of one vertex at a time, while always maintaining a k-coloring at each step. We consider two variants of graph coloring: acyclic coloring and equitable coloring, and their corresponding reconfiguration problems. A k-acylic coloring is a k-coloring where there are more than two colors used by the vertices of each cycle, and a k-equitable coloring is a k-coloring such that each color class, which is defined as the set of all vertices with a particular color, is nearly the same size as all others. We show that reconfiguration of acyclic colorings is PSPACE-hard, and that for non-bipartite graphs with chromatic number 3 there exist two k-acylic colorings $f_s$ and $f_e$ such that there is no sequence of transformations that can transform $f_s$ to $f_e$. We also consider the problem of whether two k-acylic colorings can be transformed to each other in at most $\ell$ steps, and show that it is in XP, which is the class of algorithms that run in time $O(n^{f(k)})$ for some computable function $f$ and parameter $k$, where in this case the parameter is defined to be the length of the reconfiguration sequence plus the length of the longest induced cycle. We also show that the reconfiguration of equitable colorings is PSPACE-hard and W[1]-hard with respect to the number of vertices with the same color. We give polynomial-time algorithms for Reconfiguration of Equitable Colorings when the number of colors used is two and also for paths when the number of colors used is three

Topics in graph colouring and extremal graph theory

In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let $G$ be a connected graph with $n$ vertices and maximum degree $\Delta(G)$. Let $R_k(G)$ denote the graph with vertex set all proper $k$-colourings of $G$ and two $k$-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that $R_{\Delta(G)+1}(G)$ has a unique non-trivial component with diameter $O(n^2)$. This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours $a$, $b$ of a component of the subgraph induced by vertices with colour $a$ or $b$. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all $\Delta(G)$-colourings of a graph $G$ are Kempe equivalent unless $G$ is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees

Token Jumping in minor-closed classes

Given two $k$-independent sets $I$ and $J$ of a graph $G$, one can ask if it is possible to transform the one into the other in such a way that, at any step, we replace one vertex of the current independent set by another while keeping the property of being independent. Deciding this problem, known as the Token Jumping (TJ) reconfiguration problem, is PSPACE-complete even on planar graphs. Ito et al. proved in 2014 that the problem is FPT parameterized by $k$ if the input graph is $K_{3,\ell}$-free. We prove that the result of Ito et al. can be extended to any $K_{\ell,\ell}$-free graphs. In other words, if $G$ is a $K_{\ell,\ell}$-free graph, then it is possible to decide in FPT-time if $I$ can be transformed into $J$. As a by product, the TJ-reconfiguration problem is FPT in many well-known classes of graphs such as any minor-free class