Mixing 3-colourings in bipartite graphs

Abstract

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The complexity of change

“…If k = 3, then k-Colour-Path is in P; while k-Colour-Mixing is coNP-complete (Cereceda et al [13,14]). …”

“…The proof in [13] of the coNP-completeness of 3-Colour-Mixing uses the concept of folding: given two non-adjacent vertices u and v that have a common neighbour, a fold on u and v is the identification of u and v (together with removal of any double edges produced). The results in Theorem 3.4(b) are clearly the odd ones among the list of results above.…”

“…Combining observations from [12] for non-bipartite 3-colourable graphs, with the structural characterisation of bipartite graphs that are not 3mixing in [13], gives the following. Theorem 3.6 (Cereceda et al [12,13]…”

Introduction to Reconfiguration

“…This left open the question of connectivity of bipartite graphs (χ(G) = 2) for k = 3; the yes-instances were characterized, and it was shown that the problem of deciding connectedness for bipartite graphs is coNP-complete, but restricted to planar bipartite graphs can be solved in polynomial time [78]. For k = 3, it was shown that the diameter is in O(|V(G)| 2 ) for each connected component (and that this bound is tight, as there exist configurations at distance Ω(|V(G)| 2 )), and that both reachability and shortest transformation can be solved in polynomial time [32].…”

“…So far, there are no known families where the reconfiguration graph is connected and of superpolynomial diameter; in fact, Cereceda [77,78] conjectured that for k ≥ col(G) + 2, the diameter of the reconfiguration graph is in O(|V(G)| 2 ), and showed that the conjecture is true for the values k = 1 and k = ∆(G) [77]. The results on connectivity also showed quadratic bounds on diameter, including a lower bound on diameter for chordal graphs, culminating in the proof of the conjecture for k ≥ tw(G) + 2 [85].…”