Optimally Reconfiguring List and Correspondence Colourings
Stijn Cambie, Wouter Cames van Batenburg, Daniel W. Cranston

TL;DR
This paper investigates the maximum diameter of reconfiguration graphs for list and correspondence colourings, proposing conjectures and bounds that relate these diameters to graph invariants like matching and vertex cover numbers.
Contribution
It introduces conjectures on the diameter bounds for list and correspondence colourings, proves these bounds in certain cases, and provides constructions showing the bounds are tight.
Findings
Established upper bounds for diameters in list and correspondence colourings.
Proved the conjectured bounds hold when all list sizes are at least twice the degree plus one.
Validated the conjectures for specific graph classes such as bipartite, subcubic, and bounded degree graphs.
Abstract
The reconfiguration graph for the -colourings of a graph has a vertex for each proper -colouring of , and two vertices of are adjacent precisely when those -colourings differ on a single vertex of . Much work has focused on bounding the maximum value of over all -vertex graphs . We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if is a list-assignment for a graph with for all , then . We also conjecture that if is a correspondence cover for a graph with for all , then . (Here and denote the matching number and vertex cover number of .) For…
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Taxonomy
TopicsAdvanced Graph Theory Research
