Adaptable chromatic number of graph products

An asymptotically tight bound on the adaptable chromatic number

“…Intuitively, χ a is typically much less than χ. There are constructions, however, that demonstrate that χ a can be equal to χ . These constructions rely on blowing up graphs in a way that maintains χ while increasing χ a ; they have at least χ 2 vertices but contain the χ‐clique.…”

“…This graph has maximum degree k 2 and is not k ‐colorable. To see a simple graph example, we refer the reader to , where the authors prove that , if there is a projective plane of order n or n + 1. (The maximum degree of is ( n − 1) 2 .…”

“…Planar graphs were first looked at in , and later proved that four colors were sufficient without using the Four Color Theorem. The adaptable chromatic number of a graph can be arbitrarily high for any girth , and can be as large as the chromatic number . In , graphs with χ a ( G ) = 2 and graphs with ch a ( G ) = 2 were characterized; recognizing graphs with χ a ( G ) = t for t >2 is NP‐complete .…”

The Adaptable Chromatic Number and the Chromatic Number

“…Zhou [14] proved the conjecture for every graph, showing that χ a (G) ≥ (log log χ (G)). Other work on adaptable coloring can be found in [3,5,6,8,[11][12][13]15].…”

Algorithms to approximately count and sample conforming colorings of graphs

“…Hell and Zhu [14] introduced the adaptable chromatic number in 2008. Subsequently there have been a flurry of papers deriving bounds on the adaptable chromatic number in graphs and hypergraphs, the adaptable list chromatic number, and determining when a graph G is adaptably k-choosable, where each of these is a natural generalization of the standard graph theoretic notions (see, e.g., [10,13,14,17,19]). Recently, Cygan et al [5] gave a polynomial time algorithm for finding an adapted 3-coloring given a fixed edge 3-coloring of a complete graph, resolving the so-called ''stubborn problem'' in the classification of constraint satisfaction problems [3].…”