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 Revisited

A classical theorem of Gallai states that in every graph that is critical for $k$-colorings, the vertices of degree $k - 1$ induce a tree-like graph whose blocks are either complete graphs or cycles of odd length. Borodin and, independently, Erd\H{o}s et al. provided a well-known generalization of Gallai's Theorem to list colorings, where the list at each vertex has the same number of available colors as the degree of that vertex. In this paper, we obtain an analogous result for Kempe equivalence of list colorings, partially resolving a problem of Cranston and Mahmoud.Comment: 8 pages; strengthened main result and improved expositio

Shortest Reconfiguration of Colorings Under Kempe Changes

International audienc

Kempe equivalence of colourings of cubic graphs

Given a graph G=(V,E) and a proper vertex colouring of G, a Kempe chain is a subset of V that induces a maximal connected subgraph of G in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for k≥3, all k-colourings of connected k-regular graphs that are not complete are Kempe equivalent. We address the case k=3 by showing that all 3-colourings of a connected cubic graph G are Kempe equivalent unless G is the complete graph K4 or the triangular prism

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

The a-graph coloring problem

AbstractNo proof of the 4-color conjecture reveals why it is true; the goal has not been to go beyond proving the conjecture. The standard approach involves constructing an unavoidable finite set of reducible configurations to demonstrate that a minimal counterexample cannot exist. We study the 4-color problem from a different perspective. Instead of planar triangulations, we consider near-triangulations of the plane with a face of size 4; we call any such graph an a-graph. We state an a-graph coloring problem equivalent to the 4-color problem and then derive a coloring condition that a minimal a-graph counterexample must satisfy, expressing it in terms of equivalence classes under Kempe exchanges. Through a systematic search, we discover a family of a-graphs that satisfy the coloring condition, the fundamental member of which has order 12 and includes the Birkhoff diamond as a subgraph. Higher-order members include a string of Birkhoff diamonds. However, no member has an applicable parent triangulation that is internally 6-connected, a requirement for a minimal counterexample. Our research suggests strongly that the coloring and connectivity conditions for a minimal counterexample are incompatible; infinitely many a-graphs meet one condition or the other, but we find none that meets both

Kempe Equivalent List Edge-Colorings of Planar Graphs

For a list assignment $L$ and an $L$-coloring $\varphi$, a Kempe swap in $\varphi$ is \emph{$L$-valid} if it yields another $L$-coloring. Two $L$-colorings are \emph{$L$-equivalent} if we can form one from another by a sequence of $L$-valid Kempe swaps. And a graph $G$ is \emph{$L$-swappable} if every two of its $L$-colorings are $L$-equivalent. We consider $L$-swappability of line graphs of planar graphs with large maximum degree. Let $G$ be a planar graph with $\Delta(G)\ge 9$ and let $H$ be the line graph of $G$. If $L$ is a $(\Delta(G)+1)$-assignment to $H$, then $H$ is $L$-swappable. Let $G$ be a planar graph with $\Delta(G)\ge 15$ and let $H$ be the line graph of $G$. If $L$ is a $\Delta(G)$-assignment to $H$, then $H$ is $L$-swappable. The first result is analogous to one for $L$-choosability by Borodin, which was later strengthened by Bonamy. The second result is analogous to another for $L$-choosability by Borodin, which was later strengthened by Borodin, Kostochka, and Woodall.Comment: 15 pages, 5 figures, 2 page appendix; to appear in Discrete Math (special issue in honor of Landon Rabern