Mixing Homomorphisms, Recolorings, and Extending Circular Precolorings

Abstract

This work brings together ideas of mixing graph colorings, discrete homotopy, and precoloring extension. A particular focus is circular colorings. We prove that all the (k,q)‐colorings of a graph G can be obtained by successively recoloring a single vertex provided k/q≥2col(G) along the lines of Cereceda, van den Heuvel, and Johnson's result for k‐colorings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colorings coincide. As a corollary, we obtain an Albertson‐type extension theorem for (k,q)‐precolorings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs G for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.

Introduction to Reconfiguration

“…Fixed-parameter algorithms have been found when parameterized by k + (where is the length of the reconfiguration sequence) [79,87], parameterized by k and modular-width of the input graph (and hence for cographs when parameterized by k) [75], and for shortest transformation, parameterized by k and the size of the minimum vertex cover (and hence for split graphs parameterized by k) [75]. Other variants for which reconfiguration has been studied include LIST EDGE-COLORING [36,40,100], LIST(2,1)-LABELING [37], CIRCULAR COLORING [101,102], ACYCLIC COLORING [103], and EQUITABLE COLORING [103]. The problem of k-COLORING RECONFIGURATION can also be seen as a special case of HOMOMORPHISM RECONFIGURATION [101,104] and CONSTRAINT SATISFACTION RECONFIGURATION (Section 8).…”

“…Other variants for which reconfiguration has been studied include LIST EDGE-COLORING [36,40,100], LIST(2,1)-LABELING [37], CIRCULAR COLORING [101,102], ACYCLIC COLORING [103], and EQUITABLE COLORING [103]. The problem of k-COLORING RECONFIGURATION can also be seen as a special case of HOMOMORPHISM RECONFIGURATION [101,104] and CONSTRAINT SATISFACTION RECONFIGURATION (Section 8).…”

Reconfiguration of Colourings and Dominating Sets in Graphs

“…The circular mixing number 3 of G, written m c (G), is inf{r ∈ Q : r ≥ χ c (G) and C G k,q (G) is connected whenever k/q ≥ r}. Brewster and Noel [16] obtained bounds for m c (G) and posed some interesting questions. They characterised graphs G such that C G (G) is connected; this result requires a number of definitions and we omit it here.…”

Introduction to Reconfiguration

“…Fixed-parameter algorithms have been found when parameterized by k + (where is the length of the reconfiguration sequence) [79,87], parameterized by k and modular-width of input graph (and hence for cographs when parameterized by k) [75], and for shortest transformation, parameterized by k and the size of the minimum vertex cover (and hence for split graphs parameterized by k) [75]. Other variants for which reconfiguration has been studied include LIST EDGE-COLORING [36,40,100], LIST(2,1)-LABELING [37], CIRCULAR COLORING [101,102], ACYCLIC COLORING [103], and EQUITABLE COLORING [103]. The problem of k-COLORING RECONFIGURATION can also be seen as a special case of HOMOMORPHISM RECONFIGURATION [101,104] and CONSTRAINT SATISFACTION RECONFIGURATION (Section 8).…”

“…Other variants for which reconfiguration has been studied include LIST EDGE-COLORING [36,40,100], LIST(2,1)-LABELING [37], CIRCULAR COLORING [101,102], ACYCLIC COLORING [103], and EQUITABLE COLORING [103]. The problem of k-COLORING RECONFIGURATION can also be seen as a special case of HOMOMORPHISM RECONFIGURATION [101,104] and CONSTRAINT SATISFACTION RECONFIGURATION (Section 8).…”