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

Reconfiguration of list edge-colorings in a graph

11th International Symposium, WADS 2009, Banff, Canada, August 21-23, 2009. ProceedingsWe study the problem of reconfiguring one list edge-coloring of a graph into another list edge-coloring by changing one edge color at a time, while at all times maintaining a list edge-coloring, given a list of allowed colors for each edge. First we show that this problem is PSPACE-complete, even for planar graphs of maximum degree 3 and just six colors. Then we consider the problem restricted to trees. We show that any list edge-coloring can be transformed into any other under the sufficient condition that the number of allowed colors for each edge is strictly larger than the degrees of both its endpoints. This sufficient condition is best possible in some sense. Our proof yields a polynomial-time algorithm that finds a transformation between two given list edge-colorings of a tree with n vertices using O(n [superscript 2]) recolor steps. This worst-case bound is tight: we give an infinite family of instances on paths that satisfy our sufficient condition and whose reconfiguration requires Ω(n [superscript 2]) recolor steps

A reconfigurations analogue of Brooks’ theorem.

Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless G is a complete graph or a cycle with an odd number of vertices, or k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike. We use this result to study the reconfiguration graph R k (G) of the k-colourings of G. The vertex set of R k (G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that if k ≤ Δ(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter, if k ≥ Δ(G) + 2, then R k (G) is connected and has diameter O(n 2). We complete this structural classification by settling the missing case: if k = Δ(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2). We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R k (G) by settling the case k = Δ(G) + 1. The problem is O(n 2) time solvable for k = 3, PSPACE-complete for 4 ≤ k ≤ Δ(G), O(n) time solvable for k = Δ(G) + 1, O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes)

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

A polynomial version of Cereceda's conjecture

Let k and d be such that k ≥ d + 2. Consider two k-colourings of a d-degenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length. The k-reconfiguration graph of G is the graph whose vertices are the proper k-colourings of G, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the (d + 2)-reconfiguration graph of any d-degenerate graph on n vertices is O(n 2). So far, the existence of a polynomial diameter is open even for d = 2. In this paper, we prove that the diameter of the k-reconfiguration graph of a d-degenerate graph is O(n d+1) for k ≥ d + 2. Moreover, we prove that if k ≥ 3 2 (d + 1) then the diameter of the k-reconfiguration graph is quadratic, improving the previous bound of k ≥ 2d + 1. We also show that the 5-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs

Graph Editing to a Given Neighbourhood Degree List is Fixed-Parameter Tractable

Graph editing problems have a long history and have been widely studied, with applications in biochemistry and complex network analysis. They generally ask whether an input graph can be modified by inserting and deleting vertices and edges to a graph with the desired property. We consider the problem \textsc{Graph-Edit-to-NDL} (GEN) where the goal is to modify to a graph with a given neighbourhood degree list (NDL). The NDL lists the degrees of the neighbours of vertices in a graph, and is a stronger invariant than the degree sequence, which lists the degrees of vertices. We show \textsc{Graph-Edit-to-NDL} is NP-complete and study its parameterized complexity. In parameterized complexity, a problem is said to be fixed-parameter tractable with respect to a parameter if it has a solution whose running time is a function that is polynomial in the input size but possibly superpolynomial in the parameter. Golovach and Mertzios [ICSSR, 2016] studied editing to a graph with a given degree sequence and showed the problem is fixed-parameter tractable when parameterized by $\Delta+\ell$, where $\Delta$ is the maximum degree of the input graph and $\ell$ is the number of edits. We prove \textsc{Graph-Edit-to-NDL} is fixed-parameter tractable when parameterized by $\Delta+\ell$. Furthermore, we consider a harder problem \textsc{Constrained-Graph-Edit-to-NDL} (CGEN) that imposes constraints on the NDLs of intermediate graphs produced in the sequence. We adapt our FPT algorithm for \textsc{Graph-Edit-to-NDL} to solve \textsc{Constrained-Graph-Edit-to-NDL}, which proves \textsc{Constrained-Graph-Edit-to-NDL} is also fixed-parameter tractable when parameterized by $\Delta+\ell$. Our results imply that, for graph properties that can be expressed as properties of NDLs, editing to a graph with such a property is fixed-parameter tractable when parameterized by $\Delta+\ell$. We show that this family of graph properties includes some well-known graph measures used in complex network analysis

Graph Editing to a Given Neighbourhood Degree List is Fixed-Parameter Tractable

Graph editing problems have a long history and have been widely studied, with applications in biochemistry and complex network analysis. They generally ask whether an input graph can be modified by inserting and deleting vertices and edges to a graph with the desired property. We consider the problem \textsc{Graph-Edit-to-NDL} (GEN) where the goal is to modify to a graph with a given neighbourhood degree list (NDL). The NDL lists the degrees of the neighbours of vertices in a graph, and is a stronger invariant than the degree sequence, which lists the degrees of vertices. We show \textsc{Graph-Edit-to-NDL} is NP-complete and study its parameterized complexity. In parameterized complexity, a problem is said to be fixed-parameter tractable with respect to a parameter if it has a solution whose running time is a function that is polynomial in the input size but possibly superpolynomial in the parameter. Golovach and Mertzios [ICSSR, 2016] studied editing to a graph with a given degree sequence and showed the problem is fixed-parameter tractable when parameterized by $\Delta+\ell$, where $\Delta$ is the maximum degree of the input graph and $\ell$ is the number of edits. We prove \textsc{Graph-Edit-to-NDL} is fixed-parameter tractable when parameterized by $\Delta+\ell$. Furthermore, we consider a harder problem \textsc{Constrained-Graph-Edit-to-NDL} (CGEN) that imposes constraints on the NDLs of intermediate graphs produced in the sequence. We adapt our FPT algorithm for \textsc{Graph-Edit-to-NDL} to solve \textsc{Constrained-Graph-Edit-to-NDL}, which proves \textsc{Constrained-Graph-Edit-to-NDL} is also fixed-parameter tractable when parameterized by $\Delta+\ell$. Our results imply that, for graph properties that can be expressed as properties of NDLs, editing to a graph with such a property is fixed-parameter tractable when parameterized by $\Delta+\ell$. We show that this family of graph properties includes some well-known graph measures used in complex network analysis

Reconfiguring Graph Homomorphisms on the Sphere

Given a loop-free graph $H$, the reconfiguration problem for homomorphisms to $H$ (also called $H$-colourings) asks: given two $H$-colourings $f$ of $g$ of a graph $G$, is it possible to transform $f$ into $g$ by a sequence of single-vertex colour changes such that every intermediate mapping is an $H$-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs $H$ (e.g. all $C_4$-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever $H$ is a $K_{2,3}$-free quadrangulation of the $2$-sphere (equivalently, the plane) which is not a $4$-cycle. From this result, we deduce an analogous statement for non-bipartite $K_{2,3}$-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and $4$-chromatic generalized Mycielski graphs, for which it was not. If we instead consider graphs $G$ and $H$ with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for $H$-colourings is PSPACE-complete whenever $H$ is a reflexive $K_{4}$-free triangulation of the $2$-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which $H$-Recolouring is known to be PSPACE-complete for reflexive instances.Comment: 22 pages, 9 figure