Reconfiguration graph for vertex colourings of weakly chordal graphs

Abstract

The reconfiguration graph R k (G) of the k-colourings of a graph G contains as its vertex set the k-colourings of G and two colourings are joined by an edge if they differ in colour on just one vertex of G.We show that for each k ≥ 3 there is a k-colourable weakly chordal graph G such that R k+1 (G) is disconnected. We also introduce a subclass of k-colourable weakly chordal graphs which we call k-colourable compact graphs and show that for each k-colourable compact graph G on n vertices, R k+1 (G) has diameter O(n 2 ). We show that this class contains all k-colourable co-chordal graphs and when k = 3 all 3-colourable (P 5 , P 5 , C 5 )free graphs. We also mention some open problems. *

Building a Larger Class of Graphs for Efficient Reconfiguration of Vertex Colouring

“…See also Figure 3.13. Feghali and Fiala proved that every co-chordal graph and every 3-colourable (P 5 , C 5 , co-P 5 )-free graph is compact [33]. We note that the class of OAT graphs is a strict generalization of compact graphs.…”

“…Bonamy and Bousquet [4] answered this question in the negative, using an example of Cereceda, van den Heuvel, and Johnson [15], who showed that there exists a bipartite graph G where R k+1 (G) has an isolated vertex and where k can be arbitrarily large (we will see it in Figure 3.6). Feghali and Fiala [33] also investigated this question and found an infinite family of weakly chordal graphs G where R k+1 (G) has an isolated vertex (we will see it in Figure 3.14). In the same paper, Feghali and Fiala introduced a subclass of weakly chordal graphs called compact graphs (definitions and details in Section 3.5.7).…”

“…Recently, Feghali and Fiala [33] examined a subclass of weakly chordal graphs called compact graphs defined below. A 2-pair {u, v} is a pair of non-adjacent vertices such that every chordless path between u and v has exactly two edges.…”

“…Then we can construct G from G \ x by adding back the comparable vertex x and the appropriate edges. A compact graph that is not a complete graph and that does not have a comparable vertex [33].…”

“…Feghali and Fiala [33] investigated the reconfiguration graph for the class of weakly chordal graphs. They found an infinite family {G k | k ≥ 3} of k-colourable weakly chordal graphs where R k+1 (G k ) has an isolated vertex.…”