A note on the star chromatic number

Abstract

A. Vince introduced a natural generalization of graph coloring and proved some basic facts, revealing it to be a concept of interest. His work relies on continuous methods. In this note we make some simple observations that lead to a purely combinatorial treatment. Our methods yield shorter proofs and offer further insight. The following notion is implicit in

Circular chromatic number and Mycielski construction

“…It is well known [2,12] (and also follows from Lemma 1) that c must use every colour. Assume that Let Y be a pointwise-dominating set of N G (X) with |Y | = β G (X).…”

“…Suppose G = (V, The circular chromatic number (also known as the star chromatic number) is a natural generalization of the chromatic number (note that a (k, 1)-colouring is simply a k-colouring), and has been studied extensively in the past decade, [1,2,4,6,11,12,13,14,15,16]. It is known [12] that χ(G) − 1 < χ c (G) ≤ χ(G) for any graph G, and there are graphs G with χ c (G) = χ(G) as well as graphs with χ c (G) arbitrarily close to χ(G) − 1.…”

Graph homomorphisms: structure and symmetry

“…Proposition 3.23 (Bondy, Hell [18]) Let r, s be relatively prime integers such that r < s/2. Then there exists integers m, n such that r/s < m/n, and for any…”

Circular chromatic Ramsey number

“…It is also well known that always χ c (G) > χ(G) − 1 [1,13]. Hence in fact χ(G) = ⌈χ c (G)⌉, meaning that the circular chromatic number is a refinement of the chromatic number.…”

“…When a homomorphism exists from G to K p:q , we say that G is (p, q)-colorable, so χ c (G) = inf{p/q : G is (p, q)-colorable}. If G is finite, then χ c (G) is rational and the infimum is the minimum [1,13].…”