Dichotomy for bounded degree H -colouring

• Published in: Discrete Applied Mathematics Vol. 157; no. 2; pp. 201 - 210
• Main Author: Siggers, Mark H.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 28.01.2009; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Bounded degree restriction; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Constraint satisfaction problem; Dichotomy; Exact sciences and technology; Graph colouring; Graph theory; Information retrieval. Graph; Mathematical analysis; Mathematics; Ordinary differential equations; Sciences and techniques of general use; Theoretical computing; Constraint satisfaction problem; Dichotomy; Bounded degree restriction; Graph colouring; Computer theory; constraint satisfaction problem; Cycle; Optimization; Integer; Upper bound; Maximum; Infinite graph; Bipartite graph; Combinatorics
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2008.02.003

Given a graph H , let b ( H ) be the minimum integer b , if it exists, for which H -colouring is N P -complete when restricted to instances with degree bounded by b . We show that b ( H ) exists for any non-bipartite graph. This verifies for graphs the conjecture of Feder, Hell, and Huang that any CSP that is N P -complete, is N P -complete for instances of some maximum degree. Furthermore, we show the same for all projective CSP s, and we get constant upper bounds on the parameter b for various infinite classes of graph. For example, we show that b ( H ) = 3 for any graph H with girth at least 7 in which every edge lies in a g -cycle, where g is the odd-girth of H .